MASARYK UNIVERSITY
Faculty of Science
Department of mathematics and statistics
Habilitation Thesis
Brno 2016 Petr Zem´anek
MASARYK UNIVERSITY
Faculty of Science
Department of mathematics and statistics
Discrete Symplectic Systems and
Square Summable Solutions
Habilitation Thesis
Petr Zem´anek
2010 Mathematics Subject Classification:
Primary 39A10; 39A12; 39A70; 47A06.
Secondary 34B20; 34B05; 47A20; 47B25; 47B37; 47B39.
Keywords: Discrete symplectic system; time-reversed system; linear Hamiltonian difference
system; Weyl–Titchmarsh theory; eigenvalue; M(λ)-function; Atkinson-type condition; Weyl
disk; Weyl circle; Weyl solution; square summable solution; limit point case; limit circle case; limit
circle invariance; criteria; separated endpoints; jointly varying endpoints; periodic endpoints;
definiteness condition; nonhomogeneous problem; linear relation; deficiency index; self-adjoint
extension; uniqueness; Krein–von Neumann extension.
© Petr Zem´anek, Masaryk University, 2016
Preface
A mathematical theory is not to be considered complete
until you have made it so clear that you can explain it to
the first man whom you meet on the street.
David Hilbert, see [86, pg. 438]
This habilitation thesis is submitted as an integral part of the promotion to associate
professor at Masaryk University (Faculty of Science) in the field Mathematics – Mathematical
analysis. It reflects my research from the period of 2011 to 2016, which includes the
postdoc position supported by the Program “Employment of Newly Graduated Doctors
of Science for Scientific Excellence” (grant number CZ.1.07/2.3.00/30.0009) co-financed
from European Social Fund and the state budget of the Czech Republic.
I would like to express gratitude to my collaborator, colleague, friend, and former
advisor Roman ˇSimon Hilscher for his continual support, many hours of stimulating
and inspiring discussions, and his comments to a pre-final version of the manuscript.
I am also deeply indebted to Stephen Clark for the (hopefully successful and pleasant)
collaboration and to the Department of Mathematics and Statistics (Missouri University
of Science and Technology, Rolla, USA) for hosting my visits in 2012 and 2014, where
some parts of the presented results were achieved. In addition, many thanks belong
to the heads of our scientific team, Zuzana Doˇsl´a and Ondˇrej Doˇsl´y. Finally, I highly
appreciate the financial support of the Dean of the Faculty of Science provided for the
Spring semester 2016, which enabled me to focus on this text only.
Nevertheless, this work could not be written without the unflagging support and
endless tolerance of my wife P´et’a. I also apologize to our daughter Stella that we did not
spend more time together during the first months of her life. I am extremely thankful to
both of them with
(
x2
+ 9
4 y2
+ z2
− 1
)3
− x2
z3
− 9
200 y2
z3
= 0,
r(θ) = 1 +
√
− ln
(
2 e−0.812
− e−1.42 sin2
[5 (θ−π/2)/2]
)
.
The text was typeset by using XELATEX and pictures were generated by GeoGebra.
Brno, August 2016 Petr Zem´anek
Table of contents
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Notation and auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Discrete symplectic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral
parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Spectral theory on bounded interval . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Weyl disk and Weyl circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Square summable solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Illustrating examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Chapter 3. Jointly varying endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1 Weyl–Titchmarsh theory for jointly varying endpoints . . . . . . . . . . . . 42
3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Augmented symplectic system . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Chapter 4. Invariance of limit circle case for two discrete systems . . . . . . . . . . . . . . . . 55
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Chapter 5. Polynomial and analytic dependence on spectral parameter . . . . . . . . . . 65
5.1 Preliminaries and Lagrange identity . . . . . . . . . . . . . . . . . . . . . . . 66
5.2 Special examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Weyl–Titchmarsh theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Special quadratic dependence and limit circle case . . . . . . . . . . . . . . 79
5.4.1 Results for one system . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4.2 Results for two systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
– vii –
Table of contents
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations 89
6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Definiteness condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 Nonhomogeneous problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4 Maximal and minimal linear relations . . . . . . . . . . . . . . . . . . . . . . 106
6.4.1 Linear relations and definiteness . . . . . . . . . . . . . . . . . . . . . 107
6.4.2 Orthogonal decomposition of sequence spaces . . . . . . . . . . . . . 110
6.4.3 Minimal linear relation and its deficiency indices . . . . . . . . . . . . 112
6.5 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Chapter 7. Self-adjoint extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.1 Limit point criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3 Proof of main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.4 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Appendix: Linear relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
List of author’s publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
– viii –
Chapter 1
Introduction
The approach turns out to be fruitful and successful, and leads to
the effective construction as well as the theoretical understanding
of an abundance of what we call symplectic difference scheme,
or symplectic algorithms, or simply Hamiltonian algorithms,
since they present the proper way, i.e., the Hamiltonian way for
computing Hamiltonian dynamics.
Kang Feng, see [74, pg. 18]
In this habilitation thesis we present our recent contributions to the ongoing development
of the theory of square summable solutions of discrete symplectic systems. It
is based on results, which were achieved by the author and his scientific collaborators
(S. Clark and R. ˇSimon Hilscher) during his postdoctoral research in the period 2011–2016.
The major part of them was published in papers [A13,A15,A17,A18,A21] and in a more
general setting also in papers [A16,A19,A20].
Systematic research in this area began in 2010, when M. Bohner and S. Sun in [26] and
independently (and more extensively) S. Clark and the author in [A4] investigated square
summable solutions of discrete symplectic systems with a special linear dependence on
the spectral parameter, see the beginning of Chapter 2 for more details. In the present work
we collect our results for discrete symplectic systems with general linear dependence on
the spectral parameter as well as with a polynomial or analytic dependence. However,
we emphasize that these results do not only improve the type of the dependence on
the spectral parameter, but they significantly generalize and extend the results of [26]
and [A4]. We also lay foundations of the “operator theory” for discrete symplectic
systems, which is intimately connected with the topic of square summable solutions. The
thesis consists of seven chapters and an appendix.
▶ In the next sections of this introductory chapter we summarize the used notation
and some important results from linear algebra, define discrete symplectic systems,
and show some of their special cases.
▶ In Chapter 2 we develop the limit point and limit circle classification for discrete
symplectic systems, which depend linearly on the spectral parameter. In particular,
we investigate the associated eigenvalue problem with separated boundary conditions,
the Weyl disks and Weyl circles, their limiting behavior, and properties of
square summable solutions including the precise analysis of the number of linearly
independent square summable solutions as well as some criteria for the limit point
and limit circle cases. This chapter is based on [A15].
▶ Since the theory of Chapter 2 is based on separated boundary conditions, we focus in
– 1 –
Chapter 1. Introduction
Chapter 3 on the spectral theory for discrete symplectic systems with general jointly
varying endpoints. We characterize the eigenvalues, construct the M(λ)-function
and Weyl disks, their matrix radii and centers, and discuss the number of linearly
independent square summable solutions. These results include several particular
cases, such as the periodic and antiperiodic endpoints. The method utilizes a new
transformation to separated endpoints, which is simpler and more transparent than
the one in the known literature. This chapter is based on [A13].
▶ In Chapter 4 we extend the invariance of the limit circle case to two linear discrete
systems depending linearly on spectral parameter. The main result is a discrete
analogue of the corresponding continuous time statement, which was derived by
Walker for a pair of non-hermitian linear Hamiltonian differential systems in [168].
This chapter is based on [A19].
▶ In Chapter 5 we consider discrete symplectic systems with polynomial and analytic
dependence on the spectral parameter. We derive fundamental properties of these
systems (including the Lagrange identity) and discuss their connection with systems
known in the literature. In analogy with the results of Chapter 2, we present
a construction of the Weyl disks and determine the number of linearly independent
square summable solutions. In addition, we prove the invariance of the limit circle
case for a special quadratic dependence on the spectral parameter and its extension
to the case of two (generally non-symplectic) discrete systems. We also provide several
illustrative examples, one of which contradicts the invariance of the limit circle
case for symplectic systems depending truly analytically (i.e., nonpolynomially) on
the spectral parameter. This chapter is based on [A17].
▶ In Chapter 6 we study the definiteness of the discrete symplectic system, pay an attention
to a nonhomogeneous discrete symplectic system, and introduce the minimal
and maximal linear relations associated with these systems. We also show some
fundamental properties of the corresponding deficiency indices, including a relationship
between the number of square summable solutions and the dimension
of the defect subspace. Moreover, we give a sufficient condition for the existence
of a densely defined operator associated with a discrete symplectic system. This
chapter is based on [A18].
▶ In Chapter 7 we characterize all self-adjoint extensions of the minimal linear relation.
Especially for the scalar case on a finite discrete interval we present some equivalent
forms, discuss their uniqueness, and describe the Krein–vonNeumann extension. In
addition, we establish a limit point criterion, which partially generalizes a classical
limit point criterion for the second order Sturm–Liouville difference equations. This
chapter is based on [A21].
▶ In order to make the thesis self-contained we conclude this work by a short overview
of basic definitions and some important results from the theory of linear relations,
which is utilized in Chapters 6 and 7.
We close each chapter by a section concerning bibliographical notes, in which we mention
some open problems and possible directions for our future research. For readers’ convenience,
we provide also a list of symbols, which are used throughout the thesis. Finally,
we include an overview of author’s publications.
For completeness, we point out that the results of Chapters 2–5 were further extended
to symplectic systems on time scales in [A16,A19,A20]. This generalization enables us to
– 2 –
1.1. Notation and auxiliary results
unify and compare the corresponding results for linear Hamiltonian differential systems
and discrete symplectic systems. Moreover, since some of the studied problems were
not considered for linear Hamiltonian differential systems (such as the jointly varying
endpoints or the analytic dependence on the spectral parameter), it yields even new
results for these systems.
1.1 Notation and auxiliary results
In this section we summarize the notation used through this thesis and recall several
known facts from linear algebra (see also the list of symbols on page 141).
The sets of natural numbers, integers, real and complex numbers are, respectively, denoted
by N, Z, R, and C. Moreover, N0 := N ∪ {0}. For any λ ∈ C the symbols ¯λ, re(λ), im(λ),
and δ(λ) represent, respectively, the complex conjugate of λ, the real and imaginary parts of
λ, and the sign of the imaginary part of λ i.e., δ(λ) := sgn(im(λ)). We also use the symbols
C+ and C− for the upper and lower complex half-planes, i.e., we put C+ := {λ ∈ C | δ(λ) = 1}
and C− := {λ ∈ C | δ(λ) = −1}.
Typically, all vectors and matrices are written by small and capital letters, respectively.
All matrices are considered over the field of complex numbers C. For r ∈ N we denote
the r × r identity and zero matrices by Ir and 0r. If the dimension is clear from the context,
we write only I and 0 (for simplicity, the zero vector will also be denoted by 0). For
r, s ∈ N we mean by Cr×s the space of r × s complex matrices M = (mi,j)i=1,...,r
j=1,...,s
and Cr×1 is
abbreviated as Cr. If M1, . . . , Mm ∈ Cr×r, then diag{M1, . . . , Mm} represents the block
diagonal matrix M ∈ Cmr×mr with the matrices M1, . . . , Mm on the main diagonal. For
a given matrix M ∈ Cr×s we indicate by M⊤, M, M∗, rank M, tr M, det M, M > 0, M ≥ 0,
Madj, Ker M, Ran M, dim Ran M, sprad M, im(M) := (M−M∗)/(2i), and re(M) := (M+M∗)/2,
respectively, its transpose, conjugate, conjugate transpose, rank, trace, determinant, positive
definiteness, positive semidefiniteness, adjugate (or adjoint) matrix, kernel, range (or image, i.e.,
the space spanned by the columns of M), the dimension of Ran M, spectral radius, and
Hermitian components (or real and imaginary parts, see [102, pg. 170] or [16, Fact 3.7.29]).
In addition, by Mp,q we mean the submatrix of M ∈ Cr×s consisting of the first p ≤ r rows
and of the first q ≤ s columns of the matrix M and we write only Mp in the case p = q, i.e.,
for the p-th leading principal submatrix of M. We recall that two Hermitian and positive
semidefinite matrices L, M ∈ Cr×r with L ≤ M satisfy
Ran L ⊆ Ran M and rank L ≤ rank M, (1.1)
where the equalities occur simultaneously, i.e., it holds Ran L = Ran M if and only if
rank L = rank M, see e.g. [16, Fact 8.10.2]. Furthermore, for any matrices L ∈ Cr×s,
M ∈ Cs×p, P ∈ Cr×q, and Q ∈ Cr×r we have
rank L = rank LL∗
= rank L∗
L, (1.2)
rank L + rank M − s ≤ rank LM ≤ min{rank L, rank M}, (1.3)
rank(L, P) + dim[Ran L ∩ Ran P] = rank L + rank P, (1.4)
(I − Q)−1
=
∞∑
k=0
Qk
, when sprad Q < 1, (1.5)
see e.g. [16, Corollaries 2.5.1, 2.5.3, 2.5.10 and Facts 2.11.9, 4.10.5]. In the following
statements we show two important properties of unitary matrices. The first proposition
can be found in [102, Lemma 2.1.8] and the second statement is from [108, Theorem 5.3].
– 3 –
Chapter 1. Introduction
Proposition 1.1.1. Let r ∈ N and U1, U2, · · · ∈ Cr×r be a given sequence of unitary matrices. Then
there exists a subsequence Uk1
, Uk2
, . . . such that all of the entries of Ukj
converge (as a sequences
of complex numbers) to the entries of a unitary matrix U as j → ∞.
Proposition 1.1.2. Let r ∈ N and the matrices L, M ∈ Cr×r be such that rank L = ℓ and
rank M = m. Then sup{rank LUM | U ∈ Cr×r is unitary} = min{ℓ, m}.
We also point out, see e.g. [A4, Remark 2.6], that im(M) > 0 or im(M) < 0 implies the
invertibility of the matrix M. Moreover, we write only M∗−1 instead of (M∗)−1 or
(
M−1
)∗
and similarly for parameter dependent matrices
M∗
(λ) := [M(λ)]∗
, M−1
(λ) := [M(λ)]−1
, and M∗−1
(λ) := [M∗
(λ)]−1
= [M−1
(λ)]∗
.
Finally, if we denote by the symbol S⊥ the orthogonal complement of a subspace S of an
inner product space, then the codimension of S is defined as codim S := dim S⊥ and it holds
S⊥
1
⊇ S⊥
2
for any subspaces S1 ⊆ S2. Moreover, for any M ∈ Cr×s we have
Ran M = (Ker M∗
)⊥
, (1.6)
see [16, Theorem 2.4.3].
For any matrix M = (mi,j)i,j=1,...,r ∈ Cr×r we define its entrywise H¨older norm (or ℓ1-norm)
and spectral norm, respectively, as
||M||1 :=
r∑
i=1
r∑
j=1
|mi,j | and ||M||σ := max
{ √
µ | µ is an eigenvalue of M∗
M
}
,
see [102, Section 5.6] or [16, Chapter 9]. These norms satisfy the estimates
||M||σ ≤ ||M||1 ≤ r
√
rank M × ||M||σ, (1.7)
see [16, Fact 9.8.12 (v)], and possess the submultiplicative and self-adjoint properties, i.e.,
||ML||a ≤ ||M||a × ||L||a and ||M∗ ||a = ||M||a, where a = 1 or a = σ. The spectral norm
is also unitarily invariant, i.e., ||UMV||σ = ||M||σ for any unitary matrices U, V ∈ Cr×r,
which implies that ||L||σ ≤ ||M||σ for any Hermitian matrices L, M ∈ Cr×r such that L ≤ M,
see [16, Fact 9.9.5]. Moreover, the spectral norm is the matrix norm induced by the
Euclidean vector norm on Cr, i.e., by the norm ||v||2 := (v∗v)1/2 for any v ∈ Cr, see [16,
Proposition 9.49]. In other words, the inequality
||Mv||2 ≤ ||M||σ ||v||2 (1.8)
holds true for any M ∈ Cr×r and v ∈ Cr.
A matrix M ∈ Cr×r is said to be nilpotent provided there exists m ∈ N such that Mm = 0.
The following proposition can be found in [16, Fact 3.17.9].
Proposition 1.1.3. Let L, M ∈ Cr×r be such that the matrix L is nilpotent and the matrices
commute, i.e., LM = ML. Then det(L + M) = det M.
Following [16, Chapter 4], for λ ∈ C we define the polynomial matrix M(λ) as
M(λ) := λm
Mm + λm−1
Mm−1 + · · · + λM1 + M0,
where Mm, . . . , M0 ∈ Cr×r. The matrix-valued function M(λ) is called singular if det M(λ) is
zero for all λ ∈ C, otherwise M(λ) is called nonsingular. Moreover, M(λ) is called unimodular
if det M(λ) is a nonzero constant. The latter condition is equivalent to the fact that M(λ)
is nonsingular and that M−1(λ) is also a polynomial matrix, see [16, Proposition 4.3.7].
– 4 –
1.2. Discrete symplectic systems
For M ∈ Cr×r we also define the matrix exponential exp(M) as
exp(M) :=
∞∑
j=0
1
j!
Mj
.
Then for L, M, P, Q ∈ Cr×r with L being nonsingular and PQ = QP we have
det[exp(M)] = etr M
, L exp(M)L−1
= exp(LML−1
), (1.9)
exp(P) exp(Q) = exp(Q) exp(P) = exp(P + Q), (1.10)
see [16, Corollary 11.2.4, Proposition 11.2.8(v), Corollary 11.1.6] .
If I is an interval in R, then the associated discrete interval IZ is the set of integers in I,
i.e., IZ := I ∩ Z. In particular, N = [1, ∞)Z. With N ∈ N ∪ {0, ∞} we will be interested in the
discrete intervals, which are bounded or unbounded above, i.e., IZ := [0, N + 1)Z. Then we
define I+
Z := [0, N + 1]Z with the understanding that I+
Z = IZ when N = ∞. If N is finite
we write rather [0, N]Z instead of [0, N + 1)Z.
By C(IZ)r×s we denote the space of sequences, defined on IZ, of complex r × s matrices,
where typically r ∈ {n, 2n} and 1 ≤ s ≤ 2n. Especially, we write only C(IZ)r in the case s = 1.
If M ∈ C(IZ)r×s, then M(k) := Mk for k ∈ IZ; if M(λ) ∈ C(IZ)r×s, then M(λ, k) := Mk(λ) for
k ∈ IZ. When M ∈ C(IZ)r×s and L ∈ C(IZ)s×q, then ML ∈ C(IZ)r×q, where (ML)k := Mk Lk for
all k ∈ IZ. The set C0(IZ)r×s represents the subspace of C(IZ)r×s consisting of all sequences
compactly supported in the discrete interval IZ.
The symbol means the forward difference operator acting on C(IZ)r×s, i.e., we put
( z)k := zk = zk+1 − zk. Moreover, we let zk
n
m
:= zn − zm.
Finally, the next result follows directly from [9, Theorem IV.1.1] and it concerns a sufficient
condition for the boundedness of any fundamental matrix of a recurrence relation.
Proposition 1.1.4. Let M ∈ C([0, ∞)Z)r×r be such that
∑∞
k=0 ||Mk − I||1 < ∞. Then all solutions
of the recurrence relation
uk+1 = Mk uk, k ∈ [0, ∞)Z, (1.11)
converge as k → ∞, i.e., for any fundamental matrix U ∈ C([0, ∞)Z)r×r of system (1.11) there
exists κ > 0 such that ||Uk ||1 < κ for all k ∈ [0, ∞)Z. In addition, if Mk is invertible for all
k ∈ [0, ∞)Z, then limk→∞ uk 0 for any nontrivial solution u ∈ C([0, ∞)Z)r of system (1.11).
1.2 Discrete symplectic systems
Let us define the real 2n × 2n skew-symmetric matrix
J :=
(
0 In
−In 0
)
. (1.12)
Then det J = 1 and J can be seen as a matrix analogue of the complex unit i, because
J2 = −I. Moreover, J⊤ J = I and J−1 = −J = J⊤. A matrix M ∈ C2n×2n is called Hamiltonian
if the matrix JM is Hermitian, i.e.,
M∗
J + JM = 0,
– 5 –
Chapter 1. Introduction
and it is said to be symplectic1,2 whenever
M∗
JM = J. (1.13)
The simplest examples of symplectic matrices are I2n and J. From (1.13) one easily
observes that every symplectic matrix is invertible and satisfies | det M| = 1 (in the case of
a real symplectic matrix we have even det M = 1, see e.g. [117, pg. 3] or [115, Appendix 3,
Theorem 5]). In addition, M is symplectic if and only if M−1 = −JM∗J. Therefore the set
of 2n × 2n symplectic matrices over C forms a group with respect to the standard matrix
multiplication. We also note that condition (1.13) is equivalent to
MJM∗
= J,
i.e., M is symplectic if and only if M∗ is symplectic.
A discrete symplectic system is the first order system of recurrence relations
zk+1 = Sk zk, (1.14)
where k belongs to some discrete interval IZ and S ∈ C(IZ)2n×2n with Sk being symplectic
matrices for all k ∈ IZ. These systems naturally arise in the discrete calculus of variations
and optimal theory as Jacobi systems obtained from the weak Pontryagin maximum
principle applied to the second variation of a functional, see e.g. [89–92,94,151]. Moreover,
these systems can be found also in numerical integration schemes for Hamiltonian systems
or in the theory of continued fractions, see e.g. [31, 47, 73–77, 140] and [3, Chapter 2],
respectively. The origin of a systematic treatment of discrete symplectic systems goes
back to [3], see also [4, 20]. However, some aspects of this theory can be observed
at least 30 years earlier in [9, Section 3]. In the last two decades, the theory of discrete
symplectic systems has been developed in various directions such as the Reid roundabout
theorem, see e.g. [18,24,53,87,92–94,137,138], trigonometric and hyperbolic systems, see
e.g. [5,21,22,58,59,A3], Sturmian, spectral, and oscillation theory, see e.g. [25,50,54–57,
62–65,67,68,149].
Since the matrices Sk are symplectic on IZ, it follows from their invertibility that any
initial value problem associated with system (1.14) and with an initial condition given
at an arbitrary point k0 ∈ IZ possesses a unique solution z ∈ C(I+
Z )2n. Moreover, any
fundamental matrix of system (1.14) is symplectic on IZ if and only if it is symplectic
at some index k ∈ IZ. The same property has also any fundamental matrix of the linear
Hamiltonian differential system
z′
(t) = H(t)z(t), (1.15)
where t belongs to some interval I and H(t) is a piecewise continuous Hamiltonian matrix
on I, see e.g. [115, Appendix 3, Theorem 3]. Therefore system (1.14) can be regarded as
1
The term symplectic in this context was suggested by the German mathematician Hermann Klaus Hugo
Weyl (1885–1955) in his book [173, pg. 165]: The name “complex group” formerly advocated by me in allusion
to line complexes, as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more
embarrassing through collision with the word “complex” in the connotation of complex number. I therefore propose to
replace it by the corresponding Greek adjective “symplectic”. Dickson calls the group the “Abelian linear group” in
homage to Abel who first studied it.
2
A matrix M ∈ C2n×2n
satisfying equality (1.13) is also referred as conjugate symplectic. Moreover, if
condition (1.13) is replaced by M⊤
JM = J and applied to matrices M ∈ C2n×2n
or M ∈ R2n×2n
, then it is called
complex or real symplectic, respectively, see e.g. [117]. Nevertheless, we suppress the adjective “conjugate”,
because we will consider only complex matrices and identity (1.13) throughout this thesis.
– 6 –
1.2. Discrete symplectic systems
the proper discrete counterpart of system (1.15), see also [52] and Remark 1.2.1(iv) below.
We note that the Hamiltonian property of the matrix H(t) implies the block structure
H(t) =
( A(t) B(t)
C(t) −A∗(t)
)
for some piecewise continuous n × n matrix-valued functions A(t), B(t),
and C(t) such that B(t) = B∗(t) and C(t) = C∗(t) on I. Moreover, system (1.15) can be
equivalently written as
− Jz′
(t) = H(t)z(t), (1.16)
where H(t) is Hermitian on I. For completeness we remark that some authors deal rather
with a matrix J instead of J given in (1.12), where J =
(
0 −In
In 0
)
= −J or more generally
J is any nonsingular 2n × 2n matrix satisfying J
∗
= −J, see e.g. [101] or [9, Chapter 9].
Moreover, it is also possible to replace J by a 2n×2n matrix-valued function J(t) such that
det J(t) 0 and J
∗
(t) = −J(t) for all t ∈ I, see e.g. [116].
Remark 1.2.1. In order to emphasize the importance and generality of system (1.14) we
show now that it includes several equations or systems, which have been intensively
studied in the literature. Therefore we fix the numbers n ∈ N, N ∈ N0 and, for simplicity,
divide the vector zk into two blocks of the same size and the coefficient matrix Sk of
system (1.14) into four blocks of the same size as
zk =
(
xk
uk
)
and Sk =
(
Ak Bk
Ck Dk
)
.
Then the symplecticity of the matrices Sk and S∗
k
is equivalent with the conditions
A∗
k Dk − C∗
k Bk = I = Ak D∗
k − Bk C∗
k (1.17)
and the matrices A∗
k Ck, B∗
k Dk, Ak B∗
k, Ck D∗
k are Hermitian. (1.18)
Let us also note that although in parts (i)–(iii) below we consider only finite discrete
intervals, the discussed equivalences remain valid (with appropriate modifications) for
any type of an unbounded discrete interval.
(i) Let the number m ∈ N be fixed and P[0]
∈ C([0, N]Z)n×n, P[1]
∈ C([0, N + 1]Z)n×n, . . . ,
P[m]
∈ C([0, N + m]Z)n×n be sequences of complex-valued n × n Hermitian matrices
with det P[m]
k
0 for all k ∈ [0, N + m]Z. Then the n-vector-valued Sturm–Liouville
difference equation of order 2m, i.e.,
m∑
s=0
(−1)s s
(
P[s]
k
s
yk+1−s
)
= 0, (1.19)
is equivalent to a discrete symplectic system of a special form. More specifically, if
y ∈ C([1−m, N+m+1]Z)n solves equation (1.19) on [0, N]Z, then z ∈ C([0, N+1]Z)2mn
with the components
xk :=


yk
...
r−1yk+1−r
...
m−1yk+1−m


, uk :=


∑m
s=1(− )s−1
(
P[s]
k
s yk+1−s
)
...
∑m
s=r(− )s−r
(
P[s]
k
s yk+1−s
)
...
P[m]
k
myk+1−m


(1.20)
– 7 –
Chapter 1. Introduction
solves symplectic system (1.14) on [0, N]Z, where S ∈ C([0, N]Z)2mn×2mn with the
mn × mn blocks
Ak :=


I I · · · I
0 I · · · I
...
...
...
...
0 · · · 0 I


, Dk :=


I 0 · · · · · · 0 P[0]
k
(
P[m]
k
)−1
−I I 0 · · · 0 P[1]
k
(
P[m]
k
)−1
0 −I I · · · 0 P[2]
k
(
P[m]
k
)−1
...
...
...
...
...
...
0 · · · · · · · · · −I I + P[m−1]
k
(
P[m]
k
)−1


, (1.21)
Bk :=


0 · · · 0
(
P[m]
k
)−1
...
...
...
...
0 · · · 0
(
P[m]
k
)−1


, Ck :=


P[0]
k
P[0]
k
· · · P[0]
k
0 P[1]
k
· · · P[1]
k
...
...
...
...
0 · · · 0 P[m−1]
k


. (1.22)
On the other hand, let z ∈ C([0, N + 1]Z)2mn solve system (1.14) on [0, N]Z with Sk
having the same block structure as in (1.21)–(1.22) and denote the first n components
of xk as yk for all k ∈ [0, N + 1]Z. Then according to the transformation in (1.20) we
can extend the definition of yk to the interval [1 − m, N + m + 1]Z. In particular, the
relation between the components of xk and yk applied at k = 0 yields y−1, . . . , y1−m,
while the relation between the components of uk and yk applied at k = N+1, together
with the invertibility of P[m]
k
on [N + 1, N + m]Z, yields yN+2, . . . , yN+m+1. Then we
have y ∈ C([1 − m, N + m + 1]Z)n, which satisfies equation (1.19) on [0, N]Z.
(ii) From the previous part it follows that any symplectic system (1.14) with Ak ≡ I
and det Bk 0 on the discrete interval [0, N]Z can be reduced to the second order
Sturm–Liouville difference equation
−
(
P[1]
k
yk
)
+ P[0]
k
yk+1 = 0. (1.23)
Indeed, if N ≥ 1 and we put yk := xk for all k ∈ [0, N + 1]Z, then equality (1.23) is
satisfied on [0, N − 1]Z with P[1]
k
:= B−1
k
and P[0]
k
:= Ck. Moreover, equation (1.23) is
a special case of the Jacobi equation
−
(
Pk yk + R∗
k yk+1
)
+ Qk yk+1 + Rk yk = 0, (1.24)
where k ∈ [0, N]Z, P, R ∈ C([0, N + 1]Z)n×n with matrices Pk being Hermitian on
[0, N + 1]Z and Pk + R∗
k
invertible for all k ∈ [0, N + 1]Z, and Q ∈ C([0, N]Z)n×n with
Qk being Hermitian on [0, N]Z. But also equation (1.24) can be written as a discrete
symplectic system and, under an additional assumption, vice versa. More precisely,
if y ∈ C([0, N +2]Z)n solves equation (1.24), then the pair xk := yk, k ∈ [0, N +2]Z, and
uk := Pk yk + R∗
k
yk+1, k ∈ [0, N + 1]Z, solves system (1.14) on [0, N]Z, where
Ak := (Pk + R∗
k)−1
Pk, Dk := (Pk + R∗
k + Rk + Qk)(Pk + R∗
k)−1
,
Bk := (Pk + R∗
k)−1
, Ck := Qk (Pk + R∗
k)−1
Pk − Rk (Pk + R∗
k)−1
R∗
k.
On the other hand, if N ≥ 1 and z ∈ C([0, N + 1]Z)2n solves system (1.14) with
det Bk 0 on [0, N]Z, then yk := xk, k ∈ [0, N + 1]Z, satisfies equation (1.24) for all
k ∈ [0, N − 1]Z with the coefficient matrices (observe that Pk + R∗
k
= B−1
k
)
Pk := B−1
k Ak, Rk := (I − A∗
k)B∗−1
k , Qk := (Dk − I)B−1
k + (A∗
k − I)B∗−1
k .
– 8 –
1.2. Discrete symplectic systems
(iii) Upon expanding the difference operators in (1.23) or in (1.24), we obtain special
cases of the symmetric three-term recurrence relation, i.e.,
Sk+1 yk+2 − Tk+1 yk+1 + S∗
k yk = 0, (1.25)
where we have k ∈ [0, N]Z, S ∈ C([0, N + 1]Z)n×n with det Sk 0 on [0, N + 1]Z, and
T ∈ C([1, N + 1]Z)n×n with T∗
k
= Tk on [1, N + 1]Z. In particular, equation (1.24) leads
to (1.25) with Sk = Pk + R∗
k
and Tk = Pk + Pk−1 + R∗
k−1
+ Rk−1 + Qk−1. Equation (1.25) is
also equivalent to a special discrete symplectic system. Indeed, if y ∈ C([0, N +2]Z)n
solves equation (1.25) and we put xk := yk for k ∈ [0, N + 2]Z and uk := Sk xk+1 for
k ∈ [0, N + 1]Z, then zk solves system (1.14) on [0, N]Z with the n × n blocks
Ak := 0, Bk := S−1
k , Ck := −S∗
k, Dk := Tk+1 S−1
k . (1.26)
On the other hand, if N ≥ 1 and z ∈ C([0, N + 1]Z)2n solves system (1.14) with
det Bk 0 on [0, N]Z, then yk := xk, k ∈ [0, N + 1]Z, satisfies equation (1.25) for all
k ∈ [0, N − 1]Z with the coefficient matrices
Sk := B−1
k , Tk := B−1
k Ak + Dk−1 B−1
k−1.
This follows immediately from the previous part and the relation between equations
(1.24) and (1.25).
(iv) Another extremely important example of system (1.14) is provided by the linear
Hamiltonian difference system
xk = Ak xk+1 + Bk uk, uk = Ck xk+1 − A∗
k uk, (1.27)
where k belongs to a discrete interval IZ, z ∈ C(I+
Z )2n, and A, B, C ∈ C(IZ)n×n with
the matrices Bk and Ck being Hermitian on IZ and the matrix I − Ak invertible for all
k ∈ IZ. If we denote by the superscript [s] the partial shift in the first component of
zk, i.e., z[s]
k
:=
( xk+1
uk
)
, then system (1.27) can be written as
zk = Hk z[s]
k
or equivalently − J zk = Hk z[s]
k
,
where H ∈ C(IZ)2n×2n with the matrices Hk =
( Ak Bk
Ck −A∗
k
)
being Hamiltonian for all
k ∈ IZ, while H ∈ C(IZ)2n×2n with Hk =
( −Ck A∗
k
Ak Bk
)
being Hermitian on IZ, compare
with systems (1.15) and (1.16). This system was introduced in [69,70] as a discrete
analogue of (1.15), however the invertibility of I − Ak guarantees that system (1.27)
can be written as symplectic system (1.14) with the coefficient matrix
Sk :=
(
(I − Ak)−1 (I − Ak)−1Bk
Ck (I − Ak)−1 I − A∗
k
+ Ck (I − Ak)−1Bk
)
. (1.28)
On the other hand, system (1.14) can be written as (1.27) only if Ak is invertible for
all k ∈ IZ, in which case
Ak := I − A−1
k , Bk := A−1
k Bk, Ck := Ck A−1
k . (1.29)
– 9 –
Chapter 1. Introduction
1.3 Bibliographical notes
The equivalence described in Remark 1.2.1(i) is motivated by [4, Section 3.5], [20, Remark
2], and [111, Lemma 3]. We note that, analogously to [139, Chapter 3], it seems to
be possible to write equation (1.19) as system (1.14) also without the regularity assumption
for the coefficient matrix P[m]
. A solution of this problem will appear soon. The
relation between system (1.14) and equations (1.24) and (1.25) from Remark 1.2.1(ii)–(iii)
was discussed in [150]. In [4, Section 3.6] similar transformation of equation (1.24) into
system (1.14) was derived as a consequence of a relation between equation (1.24) and system
(1.27), which requires the additional assumption det Pk 0. The connection between
systems (1.14) and (1.27) was shown in [2, Theorem 3], see also [4, Section 3.4].
– 10 –
Chapter 2
Weyl–Titchmarsh theory for
general linear dependence on
spectral parameter
A modern mathematical proof is not very different from
a modern machine, or a modern test setup: the simple
fundamental principles are hidden and almost invisible
under a mass of technical details.
Hermann Weyl, see [174, pg. 453]
In this chapter we present the theory of square summable solutions of discrete symplectic
systems in the form
zk+1(λ) = 5k(λ)zk(λ) with 5k(λ) := Sk + λVk, (Sλ)
where λ ∈ C is the spectral parameter and Sk and Vk are complex 2n×2n matrices satisfying
S∗
k JSk = J, S∗
k JVk is Hermitian, V∗
k JVk = 0, and k := JVk JS∗
k J ≥ 0. (2.1)
Here J stands for the matrix defined in (1.12) but it is also possible to use its generalization
as discussed at the end of the paragraph preceding Remark 1.2.1, see also Section 3.3. The
indices k belong to a bounded or unbounded discrete interval as will be specified later.
The dependence on λ in (Sλ) is linear, but other than that quite general. The properties
in (2.1) imply that the matrix Sk is symplectic and that the coefficient matrix 5k(λ) of (Sλ)
satisfies the symplectic-type identity
5∗
k(¯λ)J5k(λ) = J for all λ ∈ C. (2.2)
The Hermitian matrix k will play a role of a weight for the associated semi-inner product,
see (2.26) and (2.54). Throughout this thesis we also use the standard convention that by
(Sν) we refer to the system as in (Sλ) with λ replaced by ν.
Identity (2.2) shows that the matrix 5k(λ) satisfies properties, which are similar to those
of symplectic matrices, see Section 1.2. This fact also motivates the above terminology
“symplectic system”, although system (Sλ) corresponds to the discrete symplectic system
as introduced in Section 1.2 only when λ ∈ R, such as for λ = 0. On the other hand,
system (Sλ) can be viewed as a perturbation of the original symplectic system zk+1 = Sk zk,
– 11 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
i.e., system (S0), for which the fundamental properties of symplectic systems are satisfied
with appropriate (but natural) modifications. Such properties of system (Sλ) are derived in
Section 2.1. We note that system (Sλ) with λ ∈ R was already investigated in [9, Sections 3.1
and 3.3], where the weight matrix k is also obtained for this special case. We remark that
k and J correspond to Cn and −J in [9, Formula (3.3.10)]. Particularly, in [9, Section 3.3]
it was observed that
5k(λ) = Sk + λVk = (I + λJ k)Sk. (2.3)
Hence, system (Sλ) is reduced to the equivalent form
zk+1(λ) = (I + λJ k)Sk zk(λ) with ∗
k = k and k J k = 0, (2.4)
which is more convenient in some applications, e.g., when calculating the determinant
| det 5k(λ)| = det(I + λJ k) = 1, see Lemma 2.1.3. Note that in the scalar case n = 1
the properties of k in (2.4) imply that k is a real 2 × 2 matrix. Therefore, the present
chapter extends this special form of k from the scalar (and hence real) case to any
even-dimensional complex case, cf. again [9, Section 3.3].
The origin of the theory of square integrable or summable solutions goes back to the
paper of H. Weyl [172], where the second-order Sturm–Liouville differential equation
was considered and the famous Weyl alternative was proven by using a geometrical
approach3. Weyl’s results were re-proved (by using more analytical methods) and further
extended by Titchmarsh in the series of papers summarized in [165,166]. In honor of the
pioneers, this theory is usually referred as the Weyl–Titchmarsh theory. Of course, it
has been developed in many directions during the last hundred years and we do not
attempt to delineate all details of its long history and a considerable literature, see an
outstanding overview given in [71]. Rather than that we now discuss only some crucial
moments, which are closely related to the topic of this thesis. As a natural generalization
of the theory for Sturm–Liouville differential equations, Atkinson initiated the study of
the Weyl–Titchmarsh theory for the linear Hamiltonian differential system
z′
(t) = [H(t) + λW(t)]z(t) or equivalently − Jz′
(t) =
[
H(t) + λW(t)
]
z(t) (2.5)
where H(·) and W(·) are suitable Hamiltonian matrix-valued functions with −JW(t) being
positive semidefinite, see [9] and also e.g. [30,34,35,39,97–101,108–110,123,141]. As far as
we know, the first related results devoted to the second-order difference equations were
independently given in [85, 125], see also the references mentioned in connection with
equations (2.8)–(2.10) below. However the discrete Weyl–Titchmarsh theory appears to
be substantially underdeveloped in contrast to the continuous time case. Surprisingly, its
extension to discrete systems had not attracted almost any attention (except [9, Chapter 3])
until 2004, when the Weyl–Titchmarsh theory for the linear Hamiltonian difference system
(
xk
uk
)
= (Hk + λWk)
(
xk+1
uk
)
, Hk :=
(
Ak Bk
Ck −A∗
k
)
, Wk :=
(
Ek Fk
Gk −E∗
k
)
(2.6)
was established in [36] as an answer to a remark of Professor Allan Krall, see [36, pg. 152].
In (2.6) the coefficient matrices Bk, Ck and Fk, Gk are Hermitian, i.e., Hk and Wk are Hamiltonian.
Furthermore, Wk is such that −JWk ≥ 0, and the matrix Ak(λ) := (I − Ak − λEk)−1
3
Yes, it is the same Hermann Weyl as in Section 1.2, see the first footnote on page 6. Hence the topic of this
thesis can be seen as a connection of two (originally unrelated) concepts, which were significantly influenced
by H. Weyl.
– 12 –
exists for all k and λ ∈ C, which guarantees the existence of a solution of any initial value
problem associated with (2.6) in the backward time. A special case of system (2.6) with
Ek ≡ 0 (which implies that ˜Ak(λ) ≡ ˜Ak is constant in λ) was independently and more
intensively studied in [142], see also e.g. [10,122,133,158].
As already mentioned at the beginning of Chapter 1, the study of the Weyl–Titchmarsh
theory for discrete symplectic systems was initiated in [26] and [A4], where system (Sλ)
was considered in the special case when its first equation does not depend on λ. In this
case the matrices Sk, Vk, and k have necessarily the form
Sk =
(
Ak Bk
Ck Dk
)
, Vk =
(
0 0
−Wk Ak −Wk Bk
)
, k =
(
Wk 0
0 0
)
(2.7)
with Wk ∈ Cn×n being Hermitian and positive semidefinite for all k, see also [23, Remark
3(iii)]. The theory of discrete symplectic systems (Sλ) with (2.7) has been developed
in several directions. For example, the results in [25,54,56,65,66] cover the oscillation theorems,
Sturmian theory, properties of finite eigenvalues, and the Rayleigh principle. Let
us note that the form of Vk in (2.7) follows from the perturbation of the second equation in
system (S0) by the term λWk xk+1. This approach is naturally motivated by the connection
between discrete symplectic systems and any even order vector-valued Sturm–Liouville
difference equation, Jacobi equation, and symmetric three-term recurrence relation discussed
in Remark 1.2.1(i)–(iii). More specifically, system (Sλ) with the coefficients of
the form (2.7) includes equations (1.19), (1.24), and (1.25) with the term λWk yk+1 on the
right-hand side, i.e., the equations
m∑
s=0
(−1)s s
(
P[s]
k
s
yk+1−s
)
= λWk yk+1, (2.8)
−
(
Pk yk + R∗
k yk+1
)
+ Qk yk+1 + Rk yk = λWk yk+1, (2.9)
Sk+1 yk+2 − Tk+1 yk+1 + S∗
k yk = λWk yk+1, (2.10)
which were studied e.g. in [9,11,12,15,32,33,96,103,121,147,156,157,164,170] and [105,
Chapter 7], see also [39–41]. In particular, for equation (2.8) we get system (Sλ) with the
matrix Sk as in Remark 1.2.1(i) and
Vk = −
(
0 0
V[1]
k
V[2]
k
)
, V[1]
k
=


Wk · · · Wk
0 · · · 0
...
...
...
0 · · · 0


, V[2]
k
=


0 · · · 0 Wk
(
P[m]
k
)−1
0 · · · 0 0
...
...
...
...
0 · · · 0 0


, (2.11)
which yields k = diag{Wk, 0 . . . , 0} ∈ Cmn×mn. Similarly, for equation (2.8) with m = 1
and equations (2.9), (2.10) we obtain system (Sλ) with k = diag{Wk, 0}. The aim of this
chapter is to present a generalization and extension of the results in [9,26] and [A4] to the
discrete symplectic systems of the form (Sλ). Moreover, it turns out that theses results are
more general than those in [26] and [A4] theoretically and also practically. We show (see
Example 2.5.2) that the present theory applies to certain system (Sλ) with (2.7), to which
the results in [26] and [A4] cannot be used.
It is easy to see that there is (almost) no restriction on matrices W(t) and Wk in systems
(2.5) and (2.6), respectively, while the form of Vk in (2.7) is very special. This
inconsistency represents one of our motivations for a thorough study of the discrete symplectic
systems with general linear dependence on λ and their Weyl–Titchmarsh theory
– 13 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
as presented in this chapter. However even for system (Sλ) there remains one remarkable
difference, which follows immediately from the third condition in (2.1): the matrix Vk
has to be singular for all k. Moreover, it is worth noticing that there is an interesting
overlap between system (2.6) with Ek ≡ 0 and system (Sλ), see also Remark 1.2.1(iv).
More precisely, system (Sλ) can be written as a linear Hamiltonian difference system
only if the n × n left-upper block of 5k(λ) is invertible for all λ ∈ C. However, in this
instance the dependence on λ may be nonlinear and the matrix Ek may be nonzero. On
the other hand, system (2.6) can be written as system (Sλ) only if Gk (I − Ak)−1Fk ≡ 0,
see also [142, Formula (2.3)]. Without this additional assumption we obtain a discrete
symplectic system with a special quadratic dependence on λ. This observation motivates
our study of discrete symplectic systems with polynomial and analytic dependence on
λ and their Weyl–Titchmarsh theory in Chapter 5. If, in addition, also Fk ≡ 0 in (2.6)
we get system (Sλ) with the special linear dependence described in (2.7). For completeness,
we note that the study of discrete symplectic systems has also an advantage over
the approach based on the linear difference Hamiltonian systems in an easier unification
with the continuous time theory through the calculus on time scales, see [A7,A16]. Some
problems with a unification of the Weyl–Titchmarsh theory for continuous and discrete
Hamiltonian systems are discussed in [6].
This chapter is organized as follows. In Section 2.1 we present the fundamental properties
of system (Sλ) and in Section 2.2 we study the spectral theory on a bounded interval.
In the subsequent sections we focus on the Weyl–Titchmarsh theory for system (Sλ). In
Section 2.3 we introduce the corresponding Weyl disks and Weyl circles both in the regular
and singular cases. In Section 2.4 we consider the space ℓ2 of square summable
sequences with respect to the weight k and investigate the limit point and limit circle
cases. Finally, in Section 2.5 we provide several examples illustrating our theory.
2.1 Preliminaries
First, we derive some important “symplectic” properties of the matrix 5k(λ) defined
in (Sλ). Observe that (2.2) and (2.3) imply that 5k(λ) and I + λJ k are invertible with
5−1
k (λ) = −J5∗
k(¯λ)J, (I + λJ k)−1
= I − λJ k for all λ ∈ C. (2.12)
From the invertibility of 5k(λ) we obtain the (global) existence and uniqueness of solutions
of any initial value problem associated with system (Sλ). Formula (2.12) also yields the
following straightforward facts about the coefficients Sk and Vk from (2.1).
Lemma 2.1.1. Let n ∈ N be given. For any k ∈ [0, ∞)Z the following conditions are equivalent.
(i) The matrices Sk and Vk satisfy the first three conditions in (2.1), i.e., S∗
k
JSk = J, S∗
k
JVk is
Hermitian, and V∗
k
JVk = 0.
(ii) The matrix 5k(λ) in (Sλ) satisfies (2.2), i.e., 5∗
k
(¯λ)J5k(λ) = J for all λ ∈ C.
(iii) The matrices Sk and Vk satisfy Sk JS∗
k
= J and Vk JV∗
k
= 0, and Vk JS∗
k
is Hermitian.
(iv) The matrix 5k(λ) in (Sλ) satisfies 5k(λ)J5∗
k
(¯λ) = J for all λ ∈ C.
Condition (iii) in Lemma 2.1.1 implies that the matrix k is indeed Hermitian, as
required in the main assumption (2.1). This shows that if Sk and Vk are any given matrices
satisfying the first three properties in (2.1), then the matrix k := JVk JS∗
k
J is Hermitian
and, moreover, k J k = 0. The latter equality in fact characterizes the matrices Vk for
which k has this property, i.e., the matrices Vk and k determine each other. More
– 14 –
2.1. Preliminaries
precisely, if k is any 2n × 2n Hermitian matrix such that k J k = 0, then following (2.4)
we define Vk := J k Sk. It is easy to see that the second and third properties in (2.1) are in
this case satisfied.
For convenience, we summarize the notation employed throughout this chapter.
Notation 2.1.2. The numbers n ∈ N and N ∈ [0, ∞)Z are fixed and S, V, ∈ C([0, ∞)Z)2n×2n
are such that (2.1) is satisfied for all k ∈ [0, ∞)Z.
Now we focus on the determinant of the matrix 5k(λ). Let us recall that the absolute
value of the determinant of any symplectic matrix is equal to 1 as a simple consequence
of the formula (1.13).
Lemma 2.1.3. For every λ ∈ C and k ∈ [0, ∞)Z we have | det 5k(λ)| = | det Sk | = 1.
Proof. First, from the expression given in (2.3) we obtain det 5k(λ) = det(I+λJ k)×det Sk.
Since (λJ k)2 = 0, the matrix λJ k is nilpotent of degree 2. Thus, by Proposition 1.1.3
with L := λJ k and M := I, we get det(λJ k +I) = det(L+M) = det M = 1, which implies
det 5k(λ) = det Sk. Hence the statement follows from the first condition in (2.1), because
| det 5k(λ)| = | det Sk |
(2.1)
= det J = 1. ■
Remark 2.1.4. In some special cases the result of Lemma 2.1.3 can be also verified directly.
For example, when the dependence on λ is special as displayed in (2.7), we have by [66,
pg. 1232] that
5k(λ) =
(
I 0
−λWk I
)
Sk, which implies det 5k(λ) = det Sk for all λ ∈ C. (2.13)
The following statements are direct consequences of formula (2.2) and they provide
basic properties of solutions of system (Sλ) on [0, ∞)Z. Nevertheless, it is easy to see that
the results remain valid (with appropriate modifications) for solutions of system (Sλ) on
any discrete interval IZ ⊆ [0, ∞)Z.
Lemma 2.1.5 (Wronskian-type identity). Let λ ∈ C and m ∈ N be given. If the sequences
Z(λ), Z(¯λ) ∈ C([0, ∞)Z)2n×m solve systems (Sλ) and (S¯λ) on [0, ∞)Z, respectively, then
Z∗
k(¯λ)JZk(λ) = Z∗
0(¯λ)JZ0(λ) for all k ∈ [0, ∞)Z. (2.14)
Proof. Identity (2.14) follows directly from (Sλ), (S¯λ), and (2.2), because
Z∗
k+1(¯λ)JZk+1(λ) = Z∗
k(¯λ)5∗
k(¯λ)J5k(λ)Zk(λ) = Z∗
k(¯λ)JZk(λ)
for any k ∈ [0, ∞)Z. ■
Lemma 2.1.6. Let λ ∈ C and (λ) be a fundamental matrix of system (Sλ) on [0, ∞)Z such that
∗
0(¯λ)J 0(λ) = J. (2.15)
Then for any k ∈ [0, ∞)Z we have
∗
k(¯λ)J k(λ) = J, −1
k (λ) = −J ∗
k(¯λ)J, and k(λ)J ∗
k(¯λ) = J. (2.16)
Proof. The first identity in (2.16) follows from Lemma 2.1.5 and identity (2.15). The other
two identities in (2.16) follow from the first one, since k(λ) −1
k
(λ) = I = −1
k
(λ) k(λ). ■
– 15 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
One easily observes that identity (2.15) is satisfied especially when 0(λ) ≡ 0 does
not depend on λ and 0 is symplectic.
The matrix k plays a key role in the Lagrange identity for system (Sλ), which is one
of the main tools in the whole Weyl–Titchmarsh theory, see also [9, Formula (3.7.6)].
Theorem 2.1.7 (Lagrange identity). Let λ, ν ∈ C and m ∈ N be given. If the sequences
Z(λ), Z(ν) ∈ C([0, ∞)Z)2n×m solve systems (Sλ) and (Sν) on [0, ∞)Z, respectively, then for any
k ∈ [0, ∞)Z we have
[
Z∗
k(λ)JZk(ν)
]
= (¯λ − ν)Z∗
k+1(λ) k Zk+1(ν), (2.17)
Z∗
k+1(λ)JZk+1(ν) = Z∗
0(λ)JZ0(ν) + (¯λ − ν)
k∑
j=0
Z∗
j+1(λ) j Zj+1(ν). (2.18)
Proof. Let Zk(λ) and Zk(ν) satisfy (Sλ) and (Sν) on [0, ∞)Z, respectively. Then
[
Z∗
k(λ)JZk(ν)
]
= Z∗
k+1(λ)
[
J − 5∗−1
k (λ)J5−1
k (ν)
]
Zk+1(ν)
(2.12)
= Z∗
k+1(λ)
[
J + J5k(¯λ)J5∗
k(¯ν)J
]
Zk+1(ν)
(2.1)
= (¯λ − ν)Z∗
k+1(λ) k Zk+1(ν),
which shows identity (2.17). Equality (2.18) then follows from (2.17) by summation. ■
2.2 Spectral theory on bounded interval
In this section we study the spectral properties of the corresponding regular eigenvalue
problem with separated boundary conditions. Matrices describing these boundary conditions
belong to the set
:=
{
α ∈ Cn×2n
| αα∗
= I, αJα∗
= 0
}
. (2.19)
It is known e.g. in [A4, Remark 2.7] that for any α ∈ the 2n × 2n matrix (α∗, −Jα∗) is
unitary and symplectic and it satisfies
α∗
α − Jα∗
αJ = I, i.e.,
(
α∗ −Jα∗
)−1
=
(
α
αJ
)
, and Ker α = Ran Jα∗
. (2.20)
For α ∈ we denote by (λ, α) ∈ C([0, ∞)Z)2n×2n the fundamental matrix of system (Sλ)
determined by the initial condition 0(λ, α) =
(
α∗, −Jα∗
)
, i.e.,
k+1(λ) = (Sk + λVk) k(λ), k ∈ [0, ∞)Z, 0(λ) =
(
α∗ −Jα∗
)
, λ ∈ C. (2.21)
Then the initial value 0(λ, α) is unitary, symplectic, does not depend on λ, and its inverse
is −1
0
(λ, α) = ∗
0
(λ, α). However we usually suppress the dependence on α, i.e., we write
only (λ) instead of (λ, α). In addition, we need to emphasize the two “halves” of the
fundamental matrix (λ), hence we put
k(λ) =
(
Zk(λ) Zk(λ)
)
, (2.22)
where Z(λ) = Z(λ, α) ∈ C([0, ∞)Z)2n×n and Z(λ) = Z(λ, α) ∈ C([0, ∞)Z)2n×n are the 2n × n
solutions of system (Sλ) satisfying the initial conditions Z0(λ) = α∗ and Z0(λ) = −Jα∗.
– 16 –
2.2. Spectral theory on bounded interval
Definition 2.2.1. With the fundamental matrix (λ) and its blocks specified in (2.22), we
define for M ∈ Cn×n the Weyl solution4 X(λ) ∈ C([0, ∞)Z)2n×n of system (Sλ) as
Xk(λ) = Xk(λ, α, M) := k(λ)(I, M∗
)∗
= Zk(λ) + Zk(λ)M, k ∈ [0, ∞)Z. (2.23)
For the following part (including the beginning of Section 2.3) we restrict our attention
to the finite discrete interval [0, N]Z with the fixed N ∈ [0, ∞)Z as stated in Notation 2.1.2.
Then for α, β ∈ we consider the following (regular) eigenvalue problem
(Sλ), k ∈ [0, N]Z, λ ∈ C, αz0(λ) = 0, βzN+1(λ) = 0. (2.24)
Note that when α = (I, 0) = β, the boundary conditions in (2.24) reduce to the Dirichlet
boundary conditions x0 = 0 = xN+1. On the other hand, the periodic or antiperiodic
boundary conditions z0(λ) = ±zN+1(λ) cannot be obtained through any choice of the
matrices α, β ∈ . These particular cases are special examples of jointly varying endpoints,
which are investigated in Chapter 3. We recall that a number λ ∈ C is said to
be an eigenvalue of problem (2.24) if, for this particular value λ, there exists a nontrivial
solution z(λ) ∈ C([0, N + 1]Z)2n of problem (2.24). In this case, the function z(λ) is said
to be the eigenfunction corresponding to the eigenvalue λ and the dimension of all these
eigenfunctions corresponding to λ is called the geometric multiplicity of λ.
Moreover, we introduce the following definiteness assumption, called Atkinson’s condition,
compare with [9, Formula (3.7.10)]. Throughout this chapter we will distinguish
several forms of this definiteness assumption depending on how many solutions of (Sλ) is
involved. This distinction also serves as an indicator of the minimal assumptions needed
in each result, compare with Hypotheses 2.3.4 and 2.3.7 below.
Hypothesis 2.2.2 (Weak Atkinson condition – finite). For any λ ∈ CKR every column
z(λ) of the solution Z(λ) satisfies
N∑
k=0
z∗
k+1(λ) k zk+1(λ) > 0. (2.25)
Identity (2.18) and Hypothesis 2.2.2 imply the following characterization of the eigenvalues
and eigenfunctions of problem (2.24).
Theorem 2.2.3. Let α, β ∈ be given. Then the following statements hold.
(i) A number λ ∈ C is an eigenvalue of (2.24) if and only if det βZN+1(λ) = 0. In this case, the
eigenfunctions corresponding to the eigenvalue λ have the form z(λ) = Z(λ)d on [0, N+1]Z
with nonzero d ∈ Ker βZN+1(λ). Moreover, the geometric multiplicity of λ is equal to its
algebraic multiplicity, i.e., to dim Ker βZN+1(λ).
(ii) A number λ ∈ C is an eigenvalue of problem (2.24) if and only if det(−ZN+1(λ), Jβ∗) = 0.
In this case, the algebraic and geometric multiplicities of the eigenvalue λ are equal to the
value of dim Ker(−ZN+1(λ), Jβ∗).
(iii) Under Hypothesis 2.2.2, the eigenvalues of (2.24) are real and eigenfunctions corresponding
to different eigenvalues are orthogonal with respect to the semi-inner product
⟨z, ˜z⟩ ,N :=
N∑
k=0
z∗
k+1 k ˜zk+1. (2.26)
4
The symbol X stands for the Greek letter Chi (/’ki:/).
– 17 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
Proof. The proof follows standard arguments from linear algebra about eigenvalue problems
for Hermitian matrices or self-adjoint differential or difference equations. Alternatively,
see the proofs of [A4, Lemmas 3.1, 2.9 and Theorem 2.11]. ■
Next we proceed by defining the Weyl–Titchmarsh M(λ)-function for problem (2.24).
Definition 2.2.4. Let α, β ∈ . Whenever the matrix βZk(λ) is invertible for some value
λ ∈ C and k ∈ [0, N + 1]Z, we define the Weyl–Titchmarsh M(λ)-function as the n × n matrix
Mk(λ) = Mk(λ, α, β) := −[βZk(λ)]−1
βZk(λ). (2.27)
It follows from Theorem 2.2.3 that the M(λ)-function is well defined at k = N + 1 for
every λ ∈ CKR, when Hypothesis 2.2.2 holds. Now we show the “symmetry” property
of the M(λ)-function.
Lemma 2.2.5. Let α, β ∈ and λ ∈ C be given. If k ∈ [0, N + 1]Z is such that Mk(λ) and Mk(¯λ)
exist, then
M∗
k(λ) = Mk(¯λ). (2.28)
Moreover, Mk(·) is an analytic function in its argument λ.
Proof. Let k ∈ [0, N + 1]Z. By the definition of Mk(λ), the partition of k(λ) in (2.22), and
the third formula in (2.16) we have
M∗
k(λ) − Mk(¯λ) =
[
βZk(¯λ)
]−1
β k(¯λ)J ∗
k(λ)β∗
[
βZk(λ)
]∗−1
=
[
βZk(¯λ)
]−1
βJβ∗
[
βZk(λ)
]∗−1 (2.19)
= 0,
which proves identity (2.28). The analytic property of Mk(·) follows from the fact that
Zk(λ) and Zk(λ), and hence βZk(λ) and βZk(λ), are polynomials in λ. ■
Remark 2.2.6.
(i) The Weyl solution X(λ) from Definition 2.2.1 trivially satisfies the initial boundary
condition αX0(λ) = I. In addition, if βZk(λ) is invertible for some k ∈ [0, N+1]Z, then
βXk(λ) = βZk(λ)[M − Mk(λ)]. This shows that for M = Mk(λ) we have βXk(λ) = 0.
In particular, when k = N + 1 and M = MN+1(λ), the Weyl solution X(λ) satisfies the
second boundary condition in (2.24).
(ii) We also point out that the matrix P := −βJXk(λ) ∈ Cn×n, where the Weyl solution
X(λ) is defined by (2.23) with M = Mk(λ), is invertible for any β ∈ and any given
k ∈ [0, N + 1]Z, which will be a very useful fact in the proof of the next theorem.
Indeed, the calculation
n = rank
(
β
βJ
)
Xk(λ) = rank
(
0
βJXk(λ)
)
= rank P
shows that P is invertible.
In the following theorem we specify the dependence of the Weyl-Titchmarsh M(λ)function
on the matrix α determining the initial boundary condition of the fundamental
matrix (λ) = (λ, α) in (2.22). We consider the matrix Mk(λ, α, β) defined in (2.27) and
the matrix Mk(λ, γ, β) given also by (2.27) but with α replaced by γ ∈ , i.e., Mk(λ, γ, β)
is defined through the 2n × n columns of the fundamental matrix (λ, γ) which satisfies
0(λ, γ) = (γ∗, −Jγ∗). The proofs of the next theorem and its corollary follow the similar
– 18 –
2.3. Weyl disk and Weyl circle
arguments as the corresponding proofs in [A4, Lemma 3.10 and Corollary 3.11]. Note
that the assumptions of Theorem 2.2.7 and Corollary 2.2.8 below are in particular satisfied
when λ ∈ CKR, k = N + 1, and Hypothesis 2.2.2 holds.
Theorem 2.2.7. Let β ∈ and λ ∈ C. Assume that for α, γ ∈ and k ∈ [0, N + 1]Z the matrices
Mk(λ, α, β) and Mk(λ, γ, β) exist. Then we have
Mk(λ, α, β) = [αJγ∗
+ αγ∗
Mk(λ, γ, β)][αγ∗
− αJγ∗
Mk(λ, γ, β)]−1
. (2.29)
Proof. Let X(α) := X(λ, α, Mk(λ, α, β)) and X(γ) := X(λ, γ, Mk(λ, γ, β)) be the Weyl solutions
as in (2.23) corresponding to M = Mk(λ, α, β) and M = Mk(λ, γ, β), respectively. Since
βXk(α) = 0 = βXk(γ) by Remark 2.2.6(i), it follows from the third equality in (2.20) that
there exist matrices P(α), P(γ) ∈ Cn×n such that Xk(α) = Jβ∗P(α) and Xk(γ) = Jβ∗P(γ).
Moreover, the matrices P(α) and P(γ) are invertible by Remark 2.2.6(ii), and hence
Xk(α)P−1
(α) = Jβ∗
= Xk(γ)P−1
(γ), i.e., Xk(α) = Xk(γ)P with P := P−1
(γ)P(α).
By the uniqueness of solutions of system (Sλ), it follows X(α) = X(γ)P on [0, N + 1]Z, i.e.,
Xj(α) = Xj(γ)P for all j ∈ [0, N + 1]Z. (2.30)
The choice j = 0 then yields
(
I
Mk(λ, α, β)
)
= −1
0 (λ, α) 0(λ, γ)
(
I
Mk(λ, γ, β)
)
P =
(
αγ∗ − αJγ∗ Mk(λ, γ, β)
αJγ∗ + αγ∗ Mk(λ, γ, β)
)
P. (2.31)
The first row of the latter identity implies P = [αγ∗ − αJγ∗ Mk(λ, γ, β)]−1, and then the
second row of (2.31) yields identity (2.29). ■
As a consequence of (2.30) and Theorem 2.2.7 we get a formula relating the Weyl
solutions corresponding to the matrices M = Mk(λ, α, β) and M = Mk(λ, γ, β) and the
initial conditions with α, γ ∈ .
Corollary 2.2.8. Let β ∈ and λ ∈ C. Assume that for α, γ ∈ and k ∈ [0, N + 1]Z the matrices
Mk(λ, α, β) and Mk(λ, γ, β) exist. Then for all j ∈ [0, N + 1]Z we have
Xj(λ, α, Mk(λ, α, β)) = Xj(λ, γ, Mk(λ, γ, β))[αγ∗
− αJγ∗
Mk(λ, γ, β)]−1
.
2.3 Weyl disk and Weyl circle
In this section we study the properties of the Weyl disks and the Weyl circles, which are
defined through the following E(M)-function. For a given α ∈ and λ ∈ CKR we define
the matrix-valued function Ek(M) = Ek(M, λ, α) : [0, N + 1]Z × Cn×n → Cn×n as
Ek(M) := iδ(λ)X∗
k(λ, α, M)JXk(λ, α, M). (2.32)
In the abbreviated form we write E(M) = iδ(λ)X∗(λ)JX(λ). The matrix Ek(M) is Hermitian
for any k ∈ [0, N + 1]Z and M ∈ Cn×n, which can be seen from the equality (iJ)∗ = iJ.
Moreover, the Lagrange identity (Theorem 2.1.7) yields that
Ek(M) = −2δ(λ) im(M) + 2| im(λ)|
k−1∑
j=0
X∗
j+1(λ) j Xj+1(λ), k ∈ [0, N + 1]Z, (2.33)
where for k = 0 the sum is zero by definition.
– 19 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
Definition 2.3.1. Let α ∈ and λ ∈ CKR. For all k ∈ [0, N + 1]Z we define the Weyl disk
Dk(λ) = Dk(λ, α) and the Weyl circle Ck(λ) = Ck(λ, α), respectively, by
Dk(λ) :=
{
M ∈ Cn×n
| Ek(M) ≤ 0
}
, Ck(λ) :=
{
M ∈ Cn×n
| Ek(M) = 0
}
.
A natural question now arises concerning the elements of Dk(λ) and Ck(λ). For
example, from (2.33) we obtain E0(M) = −2δ(λ) im(M), which implies that the Weyl circle
C0(λ) coincides with the set of all n × n Hermitian matrices, while the interior of the Weyl
disk D0(λ) is a proper subset of the set of all invertible n × n matrices. In the following
two theorems we present characterizations of matrices M lying on the Weyl circle and in
the interior of the Weyl disk.
Theorem 2.3.2. Let α ∈ , λ ∈ CKR, k ∈ [0, N + 1]Z, and M ∈ Cn×n. The matrix M belongs
to the Weyl circle Ck(λ) if and only if there exists β ∈ such that βXk(λ) = 0. In this case
M = Mk(λ), whenever the matrix Mk(λ) defined in (2.27) exists.
Proof. Assume that M ∈ Ck(λ), i.e., Ek(M) = 0. Then for the matrix γ := X∗
k
(λ)J we get
iδ(λ)γXk(λ) = Ek(M) = 0, which implies γXk(λ) = 0 and also γJγ∗ = 0. Moreover, since
rank γ = n, we have γγ∗ > 0. The matrix β := (γγ∗)−1/2 γ satisfies βXk(λ) = 0, βJβ∗ = 0,
and ββ∗ = I. Thus β ∈ as stated in the theorem.
Conversely, assume that for a given matrix M ∈ Cn×n there exists β ∈ such that
βXk(λ) = 0. Then Xk(λ) = Jβ∗P for P := −βJXk(λ), see Remark 2.2.6(ii). It follows that
Ek(M) = iδ(λ)P∗βJβ∗P = 0, so that M ∈ Ck(λ). Finally, if Mk(λ) exists, then βZk(λ) is
invertible and βZk(λ) + βZk(λ)M = βXk(λ) = 0, i.e., M = Mk(λ). ■
Theorem 2.3.3. Let α ∈ , λ ∈ CKR, k ∈ [0, N + 1]Z, and M ∈ Cn×n. The matrix M satisfies
Ek(M) < 0 if and only if there exists β ∈ Cn×2n such that iδ(λ)βJβ∗ > 0 and βXk(λ) = 0. In this
case we have with such a matrix β that M = Mk(λ), whenever the matrix Mk(λ) exists, and β may
be chosen so that ββ∗ = I.
Proof. For M ∈ Cn×n we consider the Weyl solution X(λ) given by (2.23) with n × n blocks
φ(λ) and ψ(λ), i.e., Xj(λ) = (φ∗
j
(λ), ψ∗
j
(λ))∗ for all j ∈ [0, N + 1]Z. Assume first Ek(M) < 0.
Then the matrices φk(λ) and ψk(λ) are invertible, since for a vector d ∈ Cn such that
φk(λ)d = 0 or ψk(λ)d = 0 we have d∗Ek(M)d = iδ(λ)d∗ [φ∗
k
(λ)ψk(λ) − ψ∗
k
(λ)φk(λ)]d = 0, so
that Ek(M) < 0 implies d = 0. We put γ := (I, −φk(λ)ψ−1
k
(λ)) and then we have γXk(λ) = 0
and Ek(M) = −iδ(λ)ψ∗
k
(λ)γJγ∗ ψk(λ). Since Ek(M) < 0 and ψk(λ) is invertible, it follows
that iδ(λ)γJγ∗ > 0. Finally, the matrix β := (γγ∗)−1/2γ satisfies βXk(λ) = 0, βJβ∗ > 0, and
ββ∗ = I as required in the theorem.
Conversely, assume that for a given matrix M ∈ Cn×n there exists β = (β1, β2) ∈ Cn×2n
such that βXk(λ) = 0 and iδ(λ)βJβ∗ > 0. Since 2i im(β1 β∗
2
) = βJβ∗, we can see that the
condition iδ(λ)βJβ∗ > 0 is equivalent to im(β1 β∗
2
) > 0 for im(λ) < 0 and to im(β1 β∗
2
) < 0
for im(λ) > 0. In both cases, the positive or negative definiteness of im(β1 β∗
2
) implies the
invertibility of β1 β∗
2
, and consequently the invertibility of β1 and β2 alone. Hence, from
βXk(λ) = 0 we obtain the equality φk(λ) = −β−1
1
β2 ψk(λ), and then
Ek(M) = iδ(λ)[φ∗
k(λ)ψk(λ) − ψ∗
k(λ)φk(λ)] = −iδ(λ)ψ∗
k(λ)β−1
1 (βJβ∗
)β∗−1
1 ψk(λ). (2.34)
If ψk(λ)d = 0 for some d ∈ Cn, then φk(λ)d = −β−1
1
β2 ψk(λ)d = 0. Since rank Xk(λ) = n, it
follows that d = 0, i.e., ψk(λ) is invertible. Therefore, identity (2.34) and the assumption
iδ(λ)βJβ∗ > 0 imply Ek(M) < 0. Finally, the identity M = Mk(λ) follows with the same
argument as in the final part of the proof of Theorem 2.3.2. ■
– 20 –
2.3. Weyl disk and Weyl circle
The following result shows that the matrix δ(λ) im(M) is positive semidefinite when
M belongs to the Weyl disk Dk(λ). Under an additional Atkinson-type assumption we
also obtain that δ(λ) im(M) is positive definite.
Hypothesis 2.3.4. For a given matrix M ∈ Cn×n and λ ∈ CKR, each column z(λ) of the
Weyl solution X(λ) satisfies (2.25).
Theorem 2.3.5. Let α ∈ , λ ∈ CKR, and k ∈ [0, N + 1]Z. For every matrix M ∈ Dk(λ) we have
δ(λ) im(M) ≥ | im(λ)|
k−1∑
j=0
X∗
j+1(λ) j Xj+1(λ) ≥ 0. (2.35)
Moreover, if k = N +1 and Hypothesis 2.3.4 holds, then δ(λ) im(M) > 0 and thus M is invertible.
Proof. For a matrix M ∈ Dk(λ), inequality (2.35) follows from (2.33) via Ek(M) ≤ 0. Moreover,
under Hypothesis 2.3.4 we have δ(λ) im(M) > 0, which yields that M is invertible. ■
Since the number N ∈ [0, ∞)Z was chosen arbitrarily, see Notation 2.1.2, in the remaining
part of this section we focus on the Weyl disks when k belongs to the unbounded
interval [0, ∞)Z. The first result shows that the Weyl disks are nested with increasing k.
Theorem 2.3.6. Let α ∈ and λ ∈ CKR. Then we have Dk(λ) ⊆ Dj(λ) for every k, j ∈ [0, ∞)Z
such that k ≥ j.
Proof. Let M ∈ Dk(λ), i.e., Ek(M) ≤ 0. From identity (2.33) used at indices k and j and from
the fact ℓ ≥ 0 for all ℓ ∈ [j, k − 1]Z we get Ej(M) ≤ Ek(M) ≤ 0. Therefore, M ∈ Dj(λ). ■
Our next goal is to identify the center and the matrix radii of the Weyl disks Dk(λ) for
every λ ∈ CKR, see Theorem 2.3.8. First we analyze the structure of the Ek(M) function.
From the definition of Ek(M) in (2.32) and from (2.23) one easily derives
Ek(M) =
(
I M∗
)
Kk(λ)
(
I
M
)
, Kk(λ) := iδ(λ) ∗
k(λ)J k(λ) =
(
Fk(λ) G∗
k
(λ)
Gk(λ) Hk(λ)
)
, (2.36)
where Fk(λ), Gk(λ), Hk(λ) are the n × n matrices
Fk(λ) := iδ(λ)Z∗
k(λ)JZk(λ), Gk(λ) := iδ(λ)Z
∗
k(λ)JZk(λ), Hk(λ) := iδ(λ)Z
∗
k(λ)JZk(λ).
Since Kk(λ) is Hermitian, it follows that Fk(λ) and Hk(λ) are also Hermitian. The Lagrange
identity (Theorem 2.1.7) with ν = λ then implies
Kk(λ) = iδ(λ)J + 2| im(λ)|
k−1∑
j=0
∗
j+1(λ) j j+1(λ),
from which we get the formula
Hk(λ) = 2| im(λ)|
k−1∑
j=0
Z
∗
j+1(λ) j Zj+1(λ). (2.37)
Therefore, the following Atkinson-type condition is used in order to guarantee the invertibility
(in fact, the positive definiteness) of Hk(λ) for large k; cf. Hypothesis 5.3.2. Note
also that if Hm(λ) is invertible for some m ∈ [0, ∞)Z, then it is invertible for all k ∈ [m, ∞)Z,
because the sequence of matrices Hk(λ) is nondecreasing in k as a consequence of the
fourth condition in (2.1) and identity (2.37).
– 21 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
Hypothesis 2.3.7 (Weak Atkinson condition – infinite). There exists N0 ∈ [0, ∞)Z such
that each column z(λ) of Z(λ) satisfies inequality (2.25) with N = N0 for every λ ∈ CKR.
Under Hypothesis 2.3.7, the matrices Hk(λ) are positive definite (and hence invertible)
for all k ∈ [N0 + 1, ∞)Z. For these values of k it is then possible to represent Ek(M) as
Ek(M) = Fk(λ) − G∗
k(λ)H−1
k (λ)Gk(λ) + [G∗
k(λ)H−1
k (λ) + M∗
]Hk(λ)[H−1
k (λ)Gk(λ) + M],
see also [A4, Identity (4.11)]. By using the third identity in (2.16), it follows that the
matrices Kk(λ) defined in (2.36) satisfy the symplectic-type relation
(
Fk(λ)Gk(¯λ) − G∗
k
(λ)Fk(¯λ) Fk(λ)Hk(¯λ) − G∗
k
(λ)G∗
k
(¯λ)
Gk(λ)Gk(¯λ) − Hk(λ)Fk(¯λ) Gk(λ)Hk(¯λ) − Hk(λ)G∗
k
(¯λ)
)
= K∗
k(λ)JKk(¯λ) = −J.
This implies that
G∗
k(λ)H−1
k (λ)Gk(λ) − Fk(λ) = H−1
k (¯λ) > 0 for all k ∈ [N0 + 1, ∞)Z. (2.38)
In the following theorem we justify the terminology “disk” and “circle” for Dk(λ) and
Ck(λ), respectively. In the scalar case n = 1, the sets Dk(λ) and Ck(λ) indeed represent
a disk and a circle in the complex plane similarly as in the original paper of H. Weyl,
see [172]. For this purpose, we introduce the set U of unitary matrices in Cn×n and the set
V of contractive matrices in Cn×n, i.e.,
U :=
{
U ∈ Cn×n
| U∗
U = I
}
and V :=
{
V ∈ Cn×n
| V∗
V ≤ I
}
. (2.39)
Theorem 2.3.8. Let α ∈ , λ ∈ CKR, and suppose that Hypothesis 2.3.7 holds. Then for every
k ∈ [N0 + 1, ∞)Z the Weyl disk and Weyl circle admit the representations
Dk(λ) =
{
Pk(λ) + Rk(λ)VRk(¯λ) | V ∈ V
}
, (2.40)
Ck(λ) =
{
Pk(λ) + Rk(λ)URk(¯λ) | U ∈ U
}
, (2.41)
where the matrices Pk(λ) and Rk(λ), Rk(¯λ) are defined by
Pk(λ) := −H−1
k (λ)Gk(λ) and Rk(λ) := H−1/2
k
(λ), Rk(¯λ) := H−1/2
k
(¯λ). (2.42)
Consequently, the sets Dk(λ) are closed and convex for every k ∈ [N0 + 1, ∞)Z.
Proof. Let k ∈ [N0 + 1, ∞)Z be fixed. Identity (2.37) and Hypothesis 2.3.7 imply that the
matrices H := Hk(λ) and H := Hk(¯λ) are positive definite, so that P := Pk(λ), R := Rk(λ),
and R := Rk(¯λ) are well defined. For any matrix M ∈ Dk(λ) we then have
0 ≥ Ek(M) = F − G∗
H−1
G + (G∗
H−1
+ M∗
)H(H−1
G + M)
= −H
−1
+ (G∗
H∗−1
+ M∗
)H(H−1
G + M) = −R
2
+ (M∗
− P∗
)R−2
(M − P), (2.43)
where the equality from (2.38) was used. Identity (2.43) can be also written as
R
−1
(M∗
− P∗
)R−2
(M − P)R
−1
≤ I, i.e., V∗
V ≤ I with V := R−1
(M − P)R
−1
. (2.44)
Therefore, the above defined matrix V belongs to the set V and M = P + RVR. This
calculation can be reversed, i.e., every matrix V ∈ V gives a unique matrix M := P + RVR,
which then belongs to Dk(λ). This leads to a bijection (even a homeomorphism) between
– 22 –
2.3. Weyl disk and Weyl circle
the matrices M ∈ Dk(λ) and the matrices V ∈ V. Therefore, the Weyl disk Dk(λ) has the
representation Dk(λ) = P + RV R, as it is stated in (2.40). In a similar way, the elements of
the Weyl circle Ck(λ) are in one-to-one correspondence with the matrices V given in (2.44),
which in this case satisfy the relation V∗V = I. This means that the Weyl circle Ck(λ)
admits the representation Ck(λ) = P + RUR given in (2.41). Finally, the Weyl disk Dk(λ) is
closed and convex, because the set V has the same properties. ■
The matrix Pk(λ) in Theorem 2.3.8 is called the center of the Weyl disk or the Weyl circle,
and the matrices Rk(λ) and Rk(¯λ) are called the matrix radii of the Weyl disk or the Weyl
circle. Given (2.37), these matrices are well defined whenever Hypothesis 2.3.7 is satisfied.
Moreover, the matrices Hk(λ) are nondecreasing for k → ∞, so that the matrix radii Rk(λ)
and Rk(¯λ) are nonincreasing as k → ∞. And since Rk(λ) and Rk(¯λ) are Hermitian and
positive definite for k ∈ [N0 + 1, ∞)Z, their limits
R+(λ) := lim
k→∞
Rk(λ), R+(¯λ) := lim
k→∞
Rk(¯λ) (2.45)
exist and satisfy R+(λ) ≥ 0 and R+(¯λ) ≥ 0. Now we show that the limit of the center Pk(λ)
also exists. The proof of this fact relies on properties of the spectral matrix norm ||·||σ
recalled in Section 1.1.
Theorem 2.3.9. Let α ∈ , λ ∈ CKR, and suppose that Hypothesis 2.3.7 holds. Then the center
Pk(λ) of the Weyl disk Dk(λ) converges for k → ∞ to a limiting matrix P+(λ) ∈ Cn×n, i.e.,
P+(λ) := lim
k→∞
Pk(λ). (2.46)
Proof. The proof utilizes three main tools: the representation of the Weyl disks in Theorem
2.3.8, the convergence of the matrix radii Rk(λ) and Rk(¯λ) to R+(λ) and R+(¯λ), and the
Cauchy criterion for sequences. Let k ≥ j ∈ [N0 + 1, ∞)Z. Then Dk(λ) ⊆ Dj(λ) by Theorem
2.3.6. Hence for a matrix M ∈ Dk(λ) there exist by Theorem 2.3.8 (unique) matrices
Vk, Vj ∈ V such that
M = Pk(λ) + Rk(λ)Vk Rk(¯λ) and M = Pj(λ) + Rj(λ)Vj Rj(¯λ). (2.47)
By comparing both equalities in (2.47) we can express the matrix Vj in terms of Vk as
Vj = R−1
j (λ)[Pk(λ) − Pj(λ) + Rk(λ)Vk Rk(¯λ)]R−1
j (¯λ). (2.48)
The right-hand side of equation (2.48) defines a continuous mapping T : V → V, which
assigns to each matrix V = Vk the matrix T(V) = Vj. Since the set V is convex and compact,
the Brouwer fixed point theorem implies that the mapping T has a fixed point, i.e., there
exists a matrix V ∈ V such that T(V) = V. Going back to equation (2.48), we get from
T(V) = V the expression
Pk(λ) − Pj(λ) = [Rj(λ) − Rk(λ)]VRj(¯λ) + Rk(λ)V[Rj(¯λ) − Rk(¯λ)].
The matrices V ∈ V satisfy ||V||σ ≤ 1, so that from the above equality we obtain
||Pk(λ) − Pj(λ)||σ ≤ ||Rj(λ) − Rk(λ)||σ × ||Rj(¯λ)||σ + ||Rk(λ)||σ × ||Rj(¯λ) − Rk(¯λ)||σ. (2.49)
Since the sequences of the matrix radii R(λ), R(¯λ) ∈ C([N0 + 1, ∞)Z)n×n converge, they
are bounded in the spectral norm, i.e., there exists K > 0 such that ||Rℓ(λ)||σ ≤ K and
||Rℓ(¯λ)||σ ≤ K for all ℓ ∈ [N0 + 1, ∞)Z. Choose now an arbitrary ε > 0. The convergence
– 23 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
of Rℓ(λ) and Rℓ(¯λ) for ℓ → ∞ yields the existence of an index m ∈ [N0 + 1, ∞)Z such that
||Rj(ν) − Rk(ν)||σ < ε/(2K) for ν ∈ {λ, ¯λ} and for every k ≥ j ≥ m. From inequality (2.49)
we then get ||Pk(λ) − Pj(λ)||σ < ε for all k ≥ j ≥ m. This shows that the sequence
P(λ) ∈ C([N0 + 1, ∞)Z)n×n is a Cauchy sequence. Hence the completeness of Cn×n in the
spectral norm implies the result. ■
From Theorems 2.3.6 and 2.3.8 it follows that the Weyl disks Dk(λ) are closed, convex,
and nested with increasing k ∈ [N0+1, ∞)Z, where the number N0 is from Hypothesis 2.3.7.
Therefore, the limit of Dk(λ) as k → ∞ exists and it is closed, convex, and nonempty.
Definition 2.3.10. Let α ∈ and λ ∈ CKR. Under Hypothesis 2.3.7, we define the limiting
Weyl disk as the set
D+(λ) := lim
k→∞
Dk(λ) =
∩
k∈[N0+1,∞)Z
Dk(λ)
The matrix P+(λ) defined in (2.46) and the matrices R+(λ) and R+(¯λ) from (2.45) are called
the center and the matrix radii of the limiting Weyl disk D+(λ).
Based on Theorem 2.3.8, the limiting Weyl disk D+(λ) has the following representation.
Corollary 2.3.11. Let α ∈ , λ ∈ CKR, and suppose that Hypothesis 2.3.7 holds. Then
D+(λ) =
{
P+(λ) + R+(λ)VR+(¯λ) | V ∈ V
}
.
The next corollary shows that the Weyl solutions X(λ) corresponding to matrices
M ∈ D+(λ) have finite “norms” with respect to the weight matrices k. And conversely,
the matrices M for which the corresponding Weyl solution X(λ) satisfies the estimate
below belongs necessarily to the limiting Weyl disk D+(λ). This result illustrates the
significance of the limiting Weyl disk, because it yields a lower bound for the number of
linearly independent square summable solutions in Section 2.4.
Corollary 2.3.12. Let α ∈ , λ ∈ CKR, M ∈ Cn×n, and suppose that Hypothesis 2.3.7 holds.
Then the matrix M belongs to the limiting Weyl disk D+(λ) if and only if
∞∑
k=0
X∗
k+1(λ) k Xk+1(λ) ≤
im(M)
im(λ)
.
Proof. It follows directly from Theorem 2.3.5, when it is applied at each k ∈ [N0+1, ∞)Z. ■
Under an additional assumption on the Weyl solution X(λ), compare with Hypothesis
2.3.4, we get from Theorem 2.3.5 also an information about the positive definiteness of
the matrix δ(λ) im(M).
Hypothesis 2.3.13. There exists N1 ∈ [0, ∞)Z such that for a given matrix M ∈ Cn×n and
λ ∈ CKR, each column z(λ) of the Weyl solution X(λ) satisfies (2.25) with N = N1.
Corollary 2.3.14. Let α ∈ , λ ∈ CKR, M ∈ D+(λ), and suppose that Hypotheses 2.3.7 and 2.3.13
hold. Then δ(λ) im(M) > 0 and hence M is invertible.
Proof. The result follows from Corollary 2.3.12 and the second part of Theorem 2.3.5,
when it is applied at each k ∈ [N0 + 1, ∞)Z ∩ [N1 + 1, ∞)Z. ■
The last result of this section describes the matrices M which lie in the “interior” of
D+(λ). This statement requires a strengthened version of Hypothesis 2.3.13. This stronger
assumption guarantees that the Weyl disks are strictly nested, i.e.,
D+(λ) ⫋ Dm(λ) ⫋ Dk(λ) for all m > k large enough.
– 24 –
2.3. Weyl disk and Weyl circle
Hypothesis 2.3.15. There exists N2 ∈ [0, ∞)Z such that for all m > k ∈ [N2, ∞)Z and for
a given matrix M ∈ Cn×n and λ ∈ CKR, each column z(λ) of the Weyl solution X(λ)
satisfies the inequality
m−1∑
j=k
z∗
j+1(λ) j zj+1(λ) > 0.
Theorem 2.3.16. Let α ∈ , λ ∈ CKR, M ∈ Cn×n, and suppose that Hypotheses 2.3.7 and 2.3.15
hold. Then M ∈ D+(λ) if and only if Ek(M) < 0 for all k ∈ [N0 + 1, ∞)Z ∩ [N2 + 1, ∞)Z.
Proof. Let us start with the condition M ∈ D+(λ), which is equivalent to M ∈ Dk(λ) for
all k ∈ [N0 + 1, ∞)Z ∩ [N2 + 1, ∞)Z. With such an index k and with m > k we get from the
representation of Ek(M) and Em(M) in (2.33) that
Ek(M) = Em(M) − 2| im(λ)|
m−1∑
j=k
X∗
j+1(λ) j Xj+1(λ).
The second term on the right-hand side of the latter equality is positive due to Hypothesis
2.3.15, while the first term satisfies Em(M) ≤ 0. Thus, M ∈ D+(λ) is truly equivalent to
Ek(M) < 0 for all k ∈ [N0 + 1, ∞)Z ∩ [N2 + 1, ∞)Z. ■
Remark 2.3.17.
(i) As in [A4, Formula (4.57)], we can define a M+(λ)-function corresponding to the
matrices from the limiting Weyl disk D+(λ). In particular, for α ∈ , λ ∈ CKR and
under Hypothesis 2.3.7 we define M+(λ) ∈ D+(λ) as the limit of a subsequence of
the matrices Mk(λ, α, βk) ∈ Dk(λ), i.e.,
M+(λ) := lim
j→∞
Mkj
(λ, α, βkj
), (2.50)
where iδ(λ)βkj
Jβ∗
kj
≥ 0 and βkj
β∗
kj
= I, see also Remark 2.4.4 below. The function
M+(λ) defined in (2.50) is called a half-line Weyl–Titchmarsh M(λ)-function and it
satisfies
M∗
+(λ) = M+(¯λ) for all λ ∈ CKR.
Moreover, it is analytic on C+ and C− (and consequently it is a Herglotz function) as
a limit of uniformly bounded analytic functions Mk(λ) with λ restricted to compact
subsets of the upper and/or lower half-planes of C, see [142, Lemma 2.14].
(ii) In [26, Definition 3.4], the matrices on the limiting Weyl circle were defined as the
elements M ∈ D+(λ) for which there exists a sequence {kj}∞
j=1
such that kj → ∞ as
j → ∞ and limj→∞ Ekj
(M) = 0. By using Corollary 2.3.12 and Theorem 2.3.2, this
condition is equivalent to
∞∑
k=0
X∗
k+1(λ) k Xk+1(λ) =
im(M)
im(λ)
. (2.51)
Moreover, with the aid of the Lagrange identity (Theorem 2.1.7) we obtain for every
k ∈ [0, ∞)Z that
X∗
k+1(λ)JXk+1(λ) = 2i im(λ)
[
im(M)
im(λ)
−
k∑
j=0
X∗
j+1(λ) j Xj+1(λ)
]
. (2.52)
– 25 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
Hence one easily concludes that the sum on the right-hand side of (2.52) converges
for k → ∞ and (2.51) is satisfied if and only if limk→∞ X∗
k+1
(λ)JXk+1(λ) = 0. The latter
statement is a generalization of [26, Theorems 3.8(ii) and 3.9] and [142, Theorem 6.3]
to systems (Sλ) with a general linear dependence on λ.
2.4 Square summable solutions
In this section we extend the classification of system (Sλ) being in the limit point and
limit circle case from [A4, Section 4] to a general linear dependence on λ. In the literature
there are different (but equivalent) approaches to this issue and we essentially follow [36]
and [A4]. Hence we consider the linear space of weighted square summable sequences
(with the weight ) with entries in C2n, i.e.,
ℓ2
= ℓ2
[0, ∞)Z :=
{
z ∈ C([0, ∞)Z)2n
| ||z|| < ∞
}
, (2.53)
where
||z|| :=
√
⟨z, z⟩ and ⟨z, ˜z⟩ :=
∞∑
k=0
z∗
k+1 k ˜zk+1. (2.54)
However we point out that ℓ2 is never a Hilbert space, because || · || is only a semi-norm
and ⟨·, ·⟩ is only a semi-inner product as a consequence of the fourth condition in (2.1),
compare with (6.60). We also denote by N(λ) the linear space of all square summable
solutions of system (Sλ), i.e.,
N(λ) :=
{
z ∈ ℓ2
| z solves system (Sλ)
}
.
Just for the sake of curiosity, we note that the space of all non-square summable solutions
does not form a linear space. In the next result we show the ℓ2 -properties of the Weyl
solution X(λ) with respect to the choice of M.
Theorem 2.4.1. Let α ∈ , λ ∈ CKR, and M ∈ Cn×n. Then the columns of X(λ) form a system
of linearly independent solutions of system (Sλ). If in addition Hypothesis 2.3.7 holds and
M ∈ D+(λ), then the columns of the Weyl solution X(λ) = X(λ, α, M) are square summable, i.e.,
they belong to N(λ) and so dim N(λ) ≥ n.
Proof. Let j ∈ {1, . . . , n} and denote by z[j]
:= X(λ)ej the columns of the Weyl solution X(λ),
where ej stands for the j-th unit vector of the standard basis in Cn. If c1 z[1]
k
+ · · · + cn z[n]
k
= 0
for some k ∈ [0, ∞)Z and c1, . . . , cn ∈ C, then Xk(λ)c = 0 with c := (c1, . . . , cn)⊤. Since k(λ)
is invertible on [0, ∞)Z, it follows from (2.23) that (I, M∗)∗c = 0, i.e., c = 0 and the solutions
z[1]
, . . . , z[n]
are linearly independent. Under Hypothesis 2.3.7 the limiting Weyl disk D+(λ)
exists and then for M ∈ D+(λ) we have by Corollary 2.3.12 that
z[j] 2
=
∞∑
k=0
z
[j]∗
k+1 k z
[j]
k+1
≤ e∗
j
im(M)
im(λ)
ej < ∞ for all j ∈ {1, . . . , n}.
Therefore, in this case z[j]
∈ N(λ) for all j ∈ {1, . . . , n}. ■
As a consequence of Theorem 2.4.1 we get under Hypothesis 2.3.7 the estimate
n ≤ dim N(λ) ≤ 2n for each λ ∈ CKR. (2.55)
This fact motivates the following notions, which correspond to the Weyl dichotomy in
the scalar case (i.e., for n = 1 we have either dim N(λ) = 1 or dim N(λ) = 2).
– 26 –
2.4. Square summable solutions
Definition 2.4.2. Let λ ∈ C. System (Sλ) is said to be in the limit point case if dim N(λ) = n
and to be in the limit circle case if dim N(λ) = 2n.
Any other case, i.e., n + 1 ≤ dim N(λ) ≤ 2n − 1, is simply called intermediate. The
special cases introduced in Definition 2.4.2 are singled out because of their sui generis
characteristics. In particular, in the limit point case there exists a unique M ∈ Cn×n such
that the corresponding Weyl solution X(λ) is square summable, while in the limit circle
case the Weyl solution X(λ) is square summable for any M ∈ Cn×n and this behavior is
invariant with respect to λ, see Theorem 2.4.17.
In the following theorem we show that the Weyl disks Dk(λ) collapse to a singleton
as k → ∞ if system (Sλ) is in the limit point case. In particular, the center P+(λ) given
by (2.46) is the only matrix belonging to D+(λ). This justifies the above terminology of
being in the limit point case for system (Sλ) with dim N(λ) = n. For this particular situation
we show in the proof below that the columns of Z(λ) do not belong to N(λ). This result
is a generalization of [A4, Lemma 4.11].
Theorem 2.4.3. Let α ∈ , λ ∈ CKR, and suppose that Hypothesis 2.3.7 holds. System (Sλ) is in
the limit point case if and only if the limiting matrix radius R+(λ) = 0. In this case the limiting
Weyl disk satisfies D+(λ) = {P+(λ)} and D+(¯λ) = {P+(¯λ)}.
Proof. Assume that system (Sλ) is in the limit point case, i.e., dim N(λ) = n. Since
the columns of the fundamental matrix (λ) introduced in (2.22) span all solutions of
system (Sλ), the definition of X(λ) as Z(λ) + Z(λ)M with M ∈ D+(λ) implies that the
columns of Z(λ) and X(λ) also form a basis of all solutions of system (Sλ). Hence, from
dim N(λ) = n and Theorem 2.4.1 we conclude that the columns of Z(λ) do not belong
to N(λ). It then follows from formula (2.37) that the matrix Hk(λ) is nondecreasing for
k ∈ [0, ∞)Z without any upper bound, i.e., its eigenvalues (being real) tend to ∞. Therefore,
the function Rk(λ) has its limit at ∞ equal to zero, i.e., R+(λ) = 0. This argument can be
reversed, i.e., if R+(λ) = 0, then the eigenvalues of Hk(λ) tend to ∞ and formula (2.37)
yields that the columns of Z(λ) do not belong to ℓ2 . Since the columns of Z(λ) and Z(λ)
form a basis of all solutions of (Sλ), it then follows that dim N(λ) ≤ n. But since at the
same time dim N(λ) ≥ n by Theorem 2.4.1, we obtain dim N(λ) = n and system (Sλ) is in
the limit point case. Finally, if R+(λ) = 0 (or, equivalently, system (Sλ) is in the limit point
case), then the equality D+(λ) = {P+(λ)} follows from Corollary 2.3.11. At the same time
we get from Corollary 2.3.11 that D+(¯λ) = {P+(¯λ)}. ■
Remark 2.4.4.
(i) In the continuous time setting, i.e., for linear Hamiltonian differential systems or
Sturm–Liouville differential equations, it can happen that the limiting Weyl disk
D+(λ) is a singleton consisting, of course, of the limiting center P+(λ), but the
corresponding limiting matrix radius R+(λ) is not the zero matrix, i.e, it satisfies
rank R+(λ) ≥ 1. Such an example is constructed in [120] for the fourth order Sturm–
Liouville differential equation. Although the same behavior can be expected also
in the discrete case, a specific example is still missing, see also Remark 2.4.21.
(ii) In addition, Theorem 2.4.3 gives a simpler characterization of the half-line Weyl–
Titchmarsh M(λ)-function M+(λ) from (2.50). In particular, in the limit point case the
limit in (2.50) can be taken over all k ∈ [N0 + 1, ∞)Z without going to subsequences
and also with βk ≡ β ∈ . That is, we have in this case
M+(λ) = lim
k→∞
Mk(λ, α, β).
– 27 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
In the next part of this section we extend the result in Theorem 2.4.3 in a way which
includes the precise effect of the matrix radii R+(λ) and R+(¯λ) on the number of linearly
independent square summable solutions of system (Sλ). Specifically, we study the
relationship between the number
r(λ) := rank R+(λ), λ ∈ CKR, (2.56)
and the dimension of N(λ). The statements in Theorem 2.4.5–Corollary 2.4.9 extend the
results in [142, Section 4] from special linear Hamiltonian difference systems to discrete
symplectic systems. In addition, these results were established as new even for system (Sλ)
with the special linear dependence on λ in (2.7). At the same time, they can be regarded
as discrete time analogues of the corresponding results for linear Hamiltonian differential
systems in [108, Section 5] and [141].
From the definition of R+(λ) in (2.45) and from (2.42) one can see that the value of r(λ)
depends also on α ∈ , i.e., we should write r(λ, α) instead of r(λ) in (2.56). But, obviously,
the number of linearly independent square summable solutions of system (Sλ) does not
depend on the choice α ∈ and we will see in Theorem 2.4.8 below that also r(λ, α) is
independent of α, so that the notation in (2.56) is justified.
Theorem 2.4.5. Let α ∈ , λ ∈ CKR, and suppose that Hypothesis 2.3.7 holds. Then system (Sλ)
has at least m := n + min{r(λ), r(¯λ)} linearly independent square summable solutions, i.e., we
have dim N(λ) ≥ m.
Proof. Consider the Weyl solution X(λ) defined through the matrix M = P+(λ), i.e.,
Xk(λ) := k(λ)
(
I
P+(λ)
)
=
(
˜z[1]
k
, . . . , ˜z[n]
k
)
for all k ∈ [0, ∞)Z,
where ˜z[j]
∈ C([0, ∞)Z)2n for j ∈ {1, . . . , n}. Then, by Theorem 2.4.1, the sequences ˜z[1]
, . . . , ˜z[n]
represent n linearly independent square summable solutions of system (Sλ). We also
consider the Weyl solution X(λ) defined by the matrix M = P+(λ) + R+(λ)UR+(¯λ), i.e.,
Xk(λ) := k(λ)
(
I
M
)
=
(
ˆz[1]
k
, . . . , ˆz[n]
k
)
for all k ∈ [0, ∞)Z,
where U ∈ U is such that rank R+(λ)UR+(¯λ) = min{r(λ), r(¯λ)}, see Proposition 1.1.2. It
follows from Theorem 2.4.1 that ˆz[1]
, . . . , ˆz[n]
are also square summable solutions of (Sλ),
because by Corollary 2.3.11 the above matrix M belongs to the limiting disk D+(λ). If we
put z[j]
:= ˜z[j]
and z[n+j]
:= ˆz[j]
for j ∈ {1, . . . , n}, then
(
z[1]
k
, . . . , z[2n]
k
)
= k(λ)
(
I I
P+(λ) M
)
= k(λ)
(
I 0
P+(λ) R+(λ)UR+(¯λ)
) (
I I
0 I
)
for all k ∈ [0, ∞)Z. Since the rank of the middle matrix on the right-hand side above is
equal to m and the other two matrices are invertible, we obtain that rank
(
z[1]
k
, . . . , z[2n]
k
)
= m
as well, from which the statement follows. ■
Before we present a precise relationship between r(λ) and dim N(λ) in Theorem 2.4.8
below, we proceed with some preliminary results. In the following theorem we establish
a connection between the value of r(λ) and the asymptotic behavior of the eigenvalues
of the matrix Hk(λ) as k → ∞. For a given λ ∈ CKR we denote by µ[1]
k
≤ · · · ≤ µ[n]
k
the eigenvalues of the positive semidefinite matrix Hk(λ) arranged in the nondecreasing
order (suppressing the argument λ).
– 28 –
2.4. Square summable solutions
Theorem 2.4.6. Let α ∈ , λ ∈ CKR, and suppose that Hypothesis 2.3.7 holds. Then for
ℓ ∈ {1, . . . , n} we have r(λ) = ℓ if and only if
0 < lim
k→∞
µ
[j]
k
=: ρ[j]
< ∞, 1 ≤ j ≤ ℓ, and lim
k→∞
µ
[j]
k
= ∞, ℓ + 1 ≤ j ≤ n.
Moreover, the numbers
(
ρ[1]
)−1/2
, . . . ,
(
ρ[ℓ]
)−1/2
are the positive eigenvalues of R+(λ).
Proof. Let k ∈ [N0 + 1, ∞)Z, where N0 is from Hypothesis 2.3.7. Since Hk(λ) is Hermitian,
there exists a unitary matrix Uk such that
U∗
k Hk(λ)Uk = diag
{
µ[1]
k
, . . . , µ[n]
k
}
. (2.57)
From the definition of Rk(λ) in (2.42) we have
U∗
k Rk(λ)Uk = diag
{(
µ[1]
k
)−1/2
, . . . ,
(
µ[n]
k
)−1/2}
, (2.58)
so that
(
µ[1]
k
)−1/2
, . . . ,
(
µ[n]
k
)−1/2
are all the eigenvalues of Rk(λ). Since the set U of unitary
matrices is compact, there exists a subsequence {Ukj
}∞
j=1
which converges as j → ∞ to
a unitary matrix U+, see also Proposition 1.1.1. Hence, from (2.58) and (2.45) we get
U∗
+ R+(λ)U+ = diag
{
lim
j→∞
(
µ[1]
kj
)−1/2
, . . . , lim
j→∞
(
µ[n]
kj
)−1/2}
.
This implies that r(λ) = ℓ if and only if the limits
lim
j→∞
µ[1]
kj
= ρ[1]
, . . . , lim
j→∞
µ[ℓ]
kj
= ρ[ℓ]
and lim
j→∞
µ[ℓ+1]
kj
= · · · = lim
j→∞
µ[n]
kj
= ∞,
where ρ[1]
, . . . , ρ[ℓ]
are finite and positive. Therefore
(
ρ[1]
)−1/2
, . . . ,
(
ρ[ℓ]
)−1/2
are the positive
eigenvalues of R+(λ), while the remaining n − ℓ eigenvalues of R+(λ) are zero. ■
The following lemma will be utilized in the first part of the proof of the subsequent
Theorem 2.4.8, which is the main result of this section.
Lemma 2.4.7. Let α ∈ , λ ∈ CKR, q ∈ {0, 1, . . . , n}, and suppose that Hypothesis 2.3.7 holds.
System (Sλ) has exactly n + q linearly independent square summable solutions if and only if there
exists an n × q matrix Q with rank Q = q such that the columns of Z(λ)Q belong to ℓ2 , and
Z(λ)η ∈ ℓ2 implies η ∈ Ran Q.
Proof. Let us assume that system (Sλ) has n + q linearly independent square summable
solutions z[1]
, . . . , z[n+q]
. By Theorem 2.4.1, these solutions can be ordered so that the first
n solutions z[1]
, . . . , z[n]
correspond to the columns of the Weyl solution, which is defined
through the center P+(λ), i.e.,
Xk(λ) := k(λ)
(
I
P+(λ)
)
=
(
z[1]
k
, . . . , z[n]
k
)
for all k ∈ [0, ∞)Z. (2.59)
Then there exists a constant 2n × q matrix K such that rank K = q and
(
z[n+1]
k
, . . . , z
[n+q]
k
)
= k(λ)K for all k ∈ [0, ∞)Z.
– 29 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
If we write K = (K∗
1
, K∗
2
)∗ with n × q blocks K1, K2, then we obtain for all k ∈ [0, ∞)Z that
(
z[1]
k
, . . . , z
[n+q]
k
)
= k(λ)
(
I K1
P+(λ) K2
)
and rank
(
I K1
P+(λ) K2
)
= n + q, (2.60)
because k(λ) is invertible on [0, ∞)Z. If we put Q := K2 − P+(λ)K1 ∈ Cn×q, then
(
In K1
P+(λ) K2
) (
In −K1
0 Iq
)
=
(
In 0
P+(λ) Q
)
. (2.61)
It now follows from (2.60) that rank Q = q. In addition, from the first equality in (2.60),
(2.61), and (2.22) we get
(
z[1]
k
, . . . , z
[n+q]
k
)
(
In −K1
0 Iq
)
= k(λ)
(
In 0
P+(λ) Q
)
=
(
Xk(λ) Zk(λ)Q
)
,
which implies that the columns of Z(λ)Q belong to ℓ2 . Hence (Xk(λ), Zk(λ)Q) consists
of n + q linearly independent square summable solutions. Finally, if we have Z(λ)η ∈ ℓ2
for some η ∈ Cn, then by the previous part there exists ξ = (ξ∗
1
, ξ∗
2
)∗ ∈ Cn+q such that
Zk(λ)η =
(
Xk(λ) Zk(λ)Q
)
ξ
(2.59)
= Zk(λ)ξ1 + Zk(λ)[P+(λ)ξ1 + Qξ2].
Since k(λ) = (Zk(λ), Zk(λ)), the above equality can be written as
k(λ)
(
ξ1
P+(λ)ξ1 + Qξ2 − η
)
= 0,
from which we get ξ1 = 0 and P+(λ)ξ1 + Qξ2 − η = 0, i.e., η = Qξ2 ∈ Ran Q as required.
Conversely, assume that there exists a matrix Q ∈ Cn×q with rank Q = q such that the
columns of Z(λ)Q belong to ℓ2 and that η ∈ Ran Q whenever Z(λ)η ∈ ℓ2 . Let X(λ) be as
in (2.59). Then the equality
Tk(λ) :=
(
Xk(λ) Zk(λ)Q
)
= k(λ)
(
In 0
P+(λ) Q
)
implies that rank Tk(λ) = n + q for all k ∈ [0, ∞)Z. This shows that system (Sλ) has at least
n + q linearly independent square summable solutions, namely these are the columns of
Tk(λ). Let z ∈ N(λ) be arbitrary. We show that zk is a linear combination of the columns
of Tk(λ), i.e., we prove that zk ∈ Ran Tk(λ) for all k ∈ [0, ∞)Z. Since the matrix
(
Xk(λ) Zk(λ)
)
= k(λ)
(
I 0
P+(λ) I
)
is also a fundamental matrix of system (Sλ), there exists ζ = (ζ∗
1
, ζ∗
2
)∗ ∈ C2n such that
zk =
(
Xk(λ) Zk(λ)
)
ζ = Xk(λ)ζ1 + Zk(λ)ζ2 for all k ∈ [0, ∞)Z.
This implies that Z(λ)ζ2 = z−X(λ)ζ1 ∈ N(λ). Thus, by the current assumption, the vector
ζ2 ∈ Ran Q, i.e., ζ2 = Qυ for some vector υ ∈ Cq. It then follows that
zk = Xk(λ)ζ1 + Zk(λ)Qυ =
(
Xk(λ) Zk(λ)Q
)
(
ζ1
υ
)
∈ Ran Tk(λ) for all k ∈ [0, ∞)Z.
Therefore, system (Sλ) has exactly n + q linearly independent square summable solutions.
■
– 30 –
2.4. Square summable solutions
Now we can give an exact relation between the number of linearly independent square
summable solutions of system (Sλ) and the rank of the limiting matrix radius R+(λ) of the
limiting Weyl disk. The result below extends and makes more precise the statements in
Theorems 2.4.1 and 2.4.5.
Theorem 2.4.8. Let α ∈ and λ ∈ CKR be given, suppose that Hypothesis 2.3.7 holds, and
define r(λ) by (2.56). Then system (Sλ) has exactly n+r(λ) linearly independent square summable
solutions, i.e., dim N(λ) = n+r(λ). Furthermore, the number r(λ) is independent of the coefficient
matrix α determining the initial boundary condition in (2.24).
Proof. Since dim N(λ), i.e., the number of square summable solutions of system (Sλ), does
not depend on the choice of α, the number r(λ) also does not depend on α. Similarly as
in [141,142], the proof is divided into two parts. In the first part we derive the estimate
dim N(λ) ≤ n + r(λ), while the opposite inequality will be given in the second part of the
proof. We abbreviate r := r(λ).
Assume that there exists a number q with r < q ≤ n such that system (Sλ) has exactly
n + q square summable solutions. By Lemma 2.4.7, there exists a constant n × q matrix Q
with rank Q = q such that the columns of Z(λ)Q belong to ℓ2 and η ∈ Ran Q whenever
Z(λ)η ∈ ℓ2 . Using (2.57), for every k ∈ [N0 + 1, ∞)Z there is a unitary matrix Uk such that
Hk(λ) = Uk diag
{
µ[1]
k
, . . . , µ[n]
k
}
U∗
k. (2.62)
By Proposition 1.1.1, there exists a subsequence {kj}∞
j=1
such that kj → ∞ as j → ∞
and Ukj
→ U+, where U+ is unitary, see also the proof of Theorem 2.4.6. If we put
Kk := U∗
k
Q =
(
K[1]∗
k
, K[2]∗
k
)∗
∈ Cn×q with K[1]
k
∈ Cr×q and K[2]
k
∈ C(n−r)×q, then
K =
(
K[1]∗
, K[2]∗
)∗
:= lim
j→∞
Kkj
= U∗
+ Q with K[1]
:= lim
j→∞
K[1]
kj
and K[2]
:= lim
j→∞
K[2]
kj
.
It can be easily seen that rank K = q, because U+ is unitary and rank Q = q. Moreover,
since q > r, it follows that rank K[1]
≤ r and rank K[2]
≥ 1. Hence there exists ξ ∈ Cq such
that K[2]
ξ 0, and then for zk := Zk(λ)Qξ on [0, ∞)Z we have z ∈ ℓ2 by Lemma 2.4.7. On
the other hand, from (2.37), (2.62), and the above definition of Kk we get
||z||2
= lim
k→∞
ξ∗
Q∗
( k−1∑
j=0
Z
∗
j+1(λ) j Zj+1(λ)
)
Qξ
(2.37)
=
1
2| im(λ)|
lim
k→∞
ξ∗
Q∗
Hk(λ)Qξ
(2.62)
=
1
2| im(λ)|
lim
k→∞
ξ∗
Q∗
Uk diag
{
µ[1]
k
, . . . , µ[n]
k
}
U∗
k Qξ
=
1
2| im(λ)|
lim
k→∞
ξ∗
(
K[1]∗
k
diag
{
µ[1]
k
, . . . , µ[r]
k
}
K[1]
k
+ K[2]∗
k
diag
{
µ[r+1]
k
, . . . , µ[n]
k
}
K[2]
k
)
ξ.
By Theorem 2.4.6 (with ℓ = r) we know that the eigenvalues µ[1]
k
, . . . , µ[r]
k
have finite limits
as k → ∞, denoted by ρ[1]
, . . . , ρ[r]
, while the eigenvalues µ[r+1]
k
, . . . , µ[n]
k
tend to ∞. This
implies that
lim
j→∞
ξ∗
K[1]∗
kj
diag
{
µ[1]
kj
, . . . , µ[r]
kj
}
K[1]
kj
ξ = ξ∗
K[1]∗
diag
{
ρ[1]
, . . . , ρ[r]
}
K[1]
ξ < ∞,
lim
j→∞
ξ∗
K[2]∗
kj
diag
{
µ[r+1]
kj
, . . . , µ[n]
kj
}
K[2]
kj
ξ = ∞,
– 31 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
because K[2]
kj
ξ → K[2]
ξ 0 as j → ∞. This shows that ||z||2
= ∞, which contradicts z ∈ ℓ2 .
Thus, system (Sλ) has at most n + r linearly independent square summable solutions.
Conversely, we will show that dim N(λ) ≥ n+r by constructing n+r linearly independent
solutions of system (Sλ) from ℓ2 . By Theorem 2.4.1, we know that the columns of the
Weyl solution defined in (2.59) form n linearly independent square summable solutions of
(Sλ). For k ∈ [N0 +1, ∞)Z, let Uk ∈ Cn×n be a unitary matrix such that (2.57) and (2.62) hold.
We put Uk =
(
U[1]
k
, U[2]
k
)
with full rank blocks U[1]
k
∈ Cn×r and U[2]
k
∈ Cn×(n−r). It follows that
the dimension of kernel of U[2]∗
k
is equal to r. Hence, if ξ[1]
k
, . . . , ξ[r]
k
is an orthonormal basis
for Ker U[2]∗
k
, then the matrix Qk :=
(
ξ[1]
k
, . . . , ξ[r]
k
)
∈ Cn×r satisfies
Q∗
k Qk = Ir and U[2]∗
k
Qk = 0(n−r)×r for all k ∈ [N0 + 1, ∞)Z, (2.63)
where 0(n−r)×r means the (n − r) × r zero matrix. By the aid of Proposition 1.1.1 again,
there exist a subsequence such that Ukj
→ U+ and Qkj
→ Q+ ∈ Cn×r for j → ∞, where
the matrix U+ =
(
U[1]
+ , U[2]
+
)
is unitary, U[1]
+ ∈ Cn×r and U[2]
+ ∈ Cn×(n−r) and, by (2.63), the
matrix Q+ satisfies Q∗
+ Q+ = Ir, rank Q+ = r, and U[2]∗
+ Q+ = 0(n−r)×r. If we denote by em for
1 ≤ m ≤ r the m-th unit vector in Cr (similarly as in the proof of Theorem 2.4.1), then
Z(λ)Q+ em
2 (2.37)
=
1
2| im(λ)|
lim
k→∞
e∗
m Q∗
+ Hk(λ)Q+ em. (2.64)
Fix now k ∈ [N0 + 1, ∞)Z. Then for every kj ≥ k we have by the monotonicity of H(λ) that
e∗
m Q∗
kj
Hk(λ)Qkj
em ≤ e∗
m Q∗
kj
Hkj
(λ)Qkj
em
(2.62)
= e∗
m Q∗
kj
Ukj
diag
{
µ[1]
kj
, . . . , µ[n]
kj
}
U∗
kj
Qkj
em
(2.63)
= e∗
m Q∗
kj
U[1]
kj
diag
{
µ[1]
kj
, . . . , µ[r]
kj
}
U[1]∗
kj
Qkj
em. (2.65)
Upon taking j → ∞ in inequality (2.65) we get
e∗
m Q∗
+ Hk(λ)Q+ em ≤ e∗
m Q∗
+ U[1]
+ diag
{
ρ[1]
, . . . , ρ[r]
}
U[1]∗
+ Q+ em =: T < ∞, (2.66)
where ρ[1]
, . . . , ρ[r]
are the finite limits of the eigenvalues µ[1]
kj
, . . . , µ[r]
kj
as j → ∞, see Theorem
2.4.6. Since the estimate in (2.66) holds for every k ∈ [N0 + 1, ∞)Z, it follows from
equality (2.64) that
2| im(λ)| × Z(λ)Q+ em
2
= lim
k→∞
e∗
m Q∗
+ Hk(λ)Q+ em
(2.66)
≤ T < ∞.
This shows that the columns of Z(λ)Q+ belong to ℓ2 . Consequently, system (Sλ) has at
least n + r linearly independent square summable solutions, which are generated by the
columns of the matrix
Yk :=
(
Zk(λ) Zk(λ)
)
(
I 0n×r
P+(λ) Q+
)
= k(λ)
(
I 0n×r
P+(λ) Q+
)
. (2.67)
Since k(λ) is invertible and rank Q+ = r, it follows that the matrix Yk in (2.67) has n + r
linearly independent columns. Hence, we proved that system (Sλ) has at least n + r
linearly independent square summable solutions, which completes the proof. ■
Combining the results of Theorems 2.4.6 and 2.4.8 we get the following supplement
of Theorem 2.4.8.
– 32 –
2.4. Square summable solutions
Corollary 2.4.9. Let α ∈ , λ ∈ CKR, and suppose that Hypothesis 2.3.7 holds. Then
0 < lim
k→∞
µ
[j]
k
=: ρ[j]
< ∞, 1 ≤ j ≤ r(λ), and lim
k→∞
µ
[j]
k
= ∞, r(λ) + 1 ≤ j ≤ n,
where the numbers
(
ρ[1]
)−1/2
, . . . ,
(
ρ[r(λ)]
)−1/2
are the positive eigenvalues of R+(λ).
Moreover, yet another simple corollary follows from Theorem 2.4.8 as a counterpart
of Theorem 2.4.3.
Corollary 2.4.10. Let α ∈ , λ ∈ CKR, and suppose that Hypothesis 2.3.7 holds. Then system
(Sλ) is in the limit circle case if and only if the matrix R+(λ) is invertible, i.e., r(λ) = n.
In the remaining part of this section we present some characterizations of the two
extreme cases r(λ) = 0 (i.e., the limit point case) and r(λ) = n (i.e., the limit circle case).
We will utilize the following strengthened Atkinson-type condition, which includes both
Hypotheses 2.3.7 and 2.3.13. An alternative terminology is that system (Sλ) is definite on
the discrete interval [0, ∞)Z, see Section 6.2 for more details.
Hypothesis 2.4.11 (Strong Atkinson condition – infinite). There exists N3 ∈ [0, ∞)Z such
that each nontrivial solution z(λ) of system (Sλ) satisfies inequality (2.25) with N = N3 for
every λ ∈ CKR.
The following three results represent direct generalizations of [A4, Theorems 4.13, 4.14
and Corollary 4.15] from the special linear dependence on λ in (2.7) to the general linear
dependence on λ. The proofs of the statements in Theorems 2.4.12 and 2.4.13 follow
exactly the same ideas as in the corresponding proofs in [A4] quoted above, which
are now considered for the general linear dependence on λ. The details are therefore
omitted. Note that Theorem 2.4.12 requires the strengthened Atkinson-type condition in
Hypothesis 2.4.11, because it uses in its proof both the limiting Weyl disk D+(λ) and the
Mk(λ) functions for large k. On the other hand, in Theorem 2.4.13 we utilize the weaker
condition from Hypothesis 2.3.7, because its proof uses only the limiting Weyl disk D+(λ).
Theorem 2.4.12. Let α ∈ , λ, ν ∈ CKR, and suppose that Hypothesis 2.4.11 holds. If systems
(Sλ) and (Sν) are both in the limit point or limit circle case, then
lim
k→∞
X∗
k(λ, α, M+(λ))JXk(ν, α, M+(ν)) = 0, (2.68)
where X(λ, α, M+(λ)) ∈ C([0, ∞)Z)2n×n and X(ν, α, M+(ν)) ∈ C([0, ∞)Z)2n×n mean the Weyl
solutions of systems (Sλ) and (Sν) defined as in (2.23) through the matrices M+(λ) and M+(ν),
respectively, which are determined by the limit in (2.50).
Theorem 2.4.13. Let α ∈ , λ ∈ CKR, and suppose that Hypothesis 2.3.7 holds. Systems (Sλ)
and (S¯λ) are in the limit point case if and only if for every square summable solutions z(λ) and
˜z(¯λ) of (Sλ) and (S¯λ), respectively, we have
z∗
k(λ)J ˜zk(¯λ) = 0 for all k ∈ [0, ∞)Z. (2.69)
Corollary 2.4.14. Let α ∈ and suppose that Hypothesis 2.4.11 holds. System (Sλ) is in the
limit point case for all λ ∈ CKR if and only if for every λ, ν ∈ CKR and every square summable
solutions z(λ) and ˜z(ν) of systems (Sλ) and (Sν), respectively, we have
lim
k→∞
z∗
k(λ)J ˜zk(ν) = 0. (2.70)
Proof. If system (Sλ) is in the limit point case for every λ ∈ CKR, then the square summable
solution z(λ) must be a constant multiple of the Weyl solution X(λ), by Theorem 2.4.1.
– 33 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
Similarly, the square summable solution z(ν) is a constant multiple of X(ν). Identity (2.70)
then follows from Theorem 2.4.12. Conversely, let (2.70) be satisfied for every λ, ν ∈ CKR
and every z(λ) ∈ N(λ) and ˜z(ν) ∈ N(ν). Fix λ ∈ CKR and put ν := ¯λ. Then from
Lemma 2.1.5 we know that the value of z∗
k
(λ)J ˜zk(¯λ) is constant on k ∈ [0, ∞)Z. Hence the
limit in (2.70) implies that identity (2.69) is satisfied, and so system (Sλ) is in the limit
point case by Theorem 2.4.13. ■
From Theorems 2.4.12 and 2.1.7 we also get the following statement.
Corollary 2.4.15. Let α ∈ , λ, ν ∈ CKR, and suppose that Hypothesis 2.4.11 holds. If systems
(Sλ) and (Sν) are both in the limit point or limit circle case, then
(¯λ − ν)
∞∑
k=0
X∗
k+1(λ, α, M+(λ)) k Xk+1(ν, α, M+(ν)) = M∗
+(λ) − M+(ν), (2.71)
where the Weyl solutions X(λ, α, M+(λ)) ∈ C([0, ∞)Z)2n×n and X(ν, α, M+(ν)) ∈ C([0, ∞)Z)2n×n
are the same as in Theorem 2.4.12.
Proof. By Theorem 2.1.7, we get that the left-hand side of (2.71) is equal to the difference
lim
k→∞
X∗
k+1(λ, α, M+(λ))JXk+1(ν, α, M+(ν)) − X∗
0(λ, α, M+(λ))JX0(ν, α, M+(ν)).
While the limit above is zero by (2.68), the second term gives by the definition of the Weyl
solution in (2.23) the equality X∗
0
(λ, α, M+(λ))JX0(ν, α, M+(ν)) = M+(ν) − M∗
+(λ). ■
Remark 2.4.16. It can be shown under Hypothesis 2.3.7 that if ψk denotes the minimal
eigenvalue of the Hermitian matrix k for k ∈ [0, ∞)Z and if
∞∑
k=0
ψk = ∞, (2.72)
then system (Sλ) is in the limit point case. This fact follows in a similar way as in [154,
Theorem 5.1] by using Theorem 2.4.13. However, system (Sλ) is such that ψk = 0 for all
k ∈ [0, ∞)Z, because the matrices k are singular, see Lemma 2.1.1 and the subsequent
paragraph. Therefore, condition (2.72) can never be fulfilled in the present theory.
We conclude this section by a generalization of one half of the classical Weyl alternative,
see e.g. [171, Theorem 8.27]. More precisely, we show that if system (Sλ) is in the limit
circle case for some λ0 ∈ C (i.e., dim N(λ0) = 2n), then it is in the limit circle case for every
λ ∈ C (i.e., dim N(λ) ≡ 2n). In other words, we derive the invariance of the limit circle case,
which provides a discrete analogue of the result established in [9, Theorem 9.11.2] for
system (2.5). However we emphasize that in contrast to the latter result we do not need to
impose any additional assumptions on the coefficient matrices of system (Sλ) in the present
setting. Similar statements for the second order Sturm–Liouville difference equations and
linear Hamiltonian difference systems can be found, respectively, in [9, Theorem 5.6.1]
and [142, Theorem 5.5]. In addition, we skip the proof, because it follows immediately
from more a general result derived in Chapter 4, see Theorem 4.2.2 and Remark 4.2.4.
Theorem 2.4.17. If there exists λ0 ∈ C such that system (Sλ0
) is in the limit circle case, then
system (Sλ) is in the limit circle case for every λ ∈ C.
– 34 –
2.4. Square summable solutions
Remark 2.4.18. From the discussion concerning equations (2.8)–(2.10) in the introduction
of this chapter and from Theorem 2.4.17 we easily deduce the invariance of the limit circle
case (i.e., of the situation when all solutions are square summable with respect to the
weight W) for any even order vector-valued Sturm–Liouville difference equation, Jacobi
equation, and symmetric three term recurrence relation. Indeed, since the corresponding
system (Sλ) is such that k = diag{Wk, 0, . . . , 0} or k = diag{Wk, 0}, we get immediately
the equality y∗
k+1
(λ)Wk yk+1(λ) = z∗
k
(λ) k zk+1(λ) for all k ∈ [0, ∞)Z. Therefore a solution
y(λ) ∈ C([0, ∞)Z)n of equations (2.8) or (2.9) or (2.10) satisfies
∑∞
k=0 y∗
k+1
(λ)Wk yk+1(λ) < ∞
if and only if the corresponding z(λ) belongs to ℓ2 with respect to the weight specified
earlier, see also Remark 4.2.7.
As a direct consequence of Theorem 2.4.17 we obtain the following criterion for the
limit circle case. This result corresponds to [9, Theorem 5.8.1] and [134, Theorem 6.3] for
the second order Sturm–Liouville difference equations and linear Hamiltonian difference
systems. Again we skip the proof, because the statement follows from Corollary 4.2.3 and
Remark 4.2.4. We note that the matrix norm ||·||1 used in (2.73) below can be replaced by
any other matrix norm because of their equivalence.
Corollary 2.4.19. Assume that
∞∑
k=0
||Sk − I||1 < ∞ and
∞∑
k=0
|| k ||1 < ∞. (2.73)
Then system (Sλ) is in the limit circle case for all λ ∈ C.
Upon combining Theorem 2.4.17 and Corollary 2.4.10 we get the following result.
Corollary 2.4.20. Assume that Hypothesis 2.3.7 holds and that λ0 ∈ C is such that (Sλ0
) is in the
limit circle case. Then r(λ) = n for all λ ∈ CKR.
Remark 2.4.21. Under the assumptions of Corollary 2.4.20 we can deduce that the value of
r(λ) is constant and equal to n on CKR. This observation gives rise to two additional and
very natural questions. Are the values of r(λ) and r(¯λ) constant on some subsets of C in
general, especially on the upper and lower half-planes C+ and C−? And if r(λ) and r(¯λ) are
constant on C+ and C−, do they satisfy r(λ) = r(¯λ) on CKR? Of course, these questions can
be formulated also for the numbers dim N(λ) and dim N(¯λ). The first answer is positive
as discussed in Remark 6.4.15. The answer to the second question is positive in the limit
circle case under the assumptions stated in Corollary 2.4.20, as well as in the limit point
case under analogous assumptions as in Theorem 7.1.1 from Chapter 7. Moreover, if the
matrices Sk and Vk are real-valued for all k ∈ [0, ∞)Z, then it follows immediately that
z(λ) ∈ C([0, ∞)Z)2n solves system (Sλ) if and only if z(λ) solves (S¯λ), i.e., zk(¯λ) = zk(λ) for all
k ∈ [0, ∞)Z. Thus, in that case we have dim N(λ) = dim N(¯λ), i.e., r(λ) = r(¯λ). However, in
other situations we conjecture that the answer can be negative similarly as it was shown
in the continuous time case by the example from [120], which was already mentioned
in Remark 2.4.4(i). More specifically, in the latter example the fourth order differential
equation with three and two linearly independent square integrable solutions in C+ and
C−, respectively, was constructed.
In the scalar case (i.e., n = 1) the estimate in (2.55) implies that dim N(λ) ∈ {1, 2}. In
this case we derive from Theorem 2.4.17 its limit point counterpart for λ ∈ CKR, i.e., the
second part of the Weyl alternative.
Corollary 2.4.22. Let n = 1 and assume that Hypothesis 2.3.7 holds. If there λ0 ∈ C such that
system (Sλ0
) is in the limit point case, then system (Sλ) is in the limit point case for any λ ∈ CKR.
– 35 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
Proof. Let system (Sλ0
) be in the limit point case and assume that there exists λ1 ∈ CKR
such that system (Sλ1
) is not in the limit point case. Then, by n = 1 and the estimate
in (2.55), we know that system (Sλ1
) is in the limit circle case. But from Theorem 2.4.17
(applied with λ0 = λ1) we obtain that system (Sλ) is in the limit circle case for every λ ∈ C,
which contradicts the original assumption that system (Sλ0
) is in the limit point case. ■
Upon combining Theorems 2.4.17 and 2.4.22 we get the scalar symplectic analogue of
the Weyl alternative for system (Sλ).
Corollary 2.4.23 (Weyl alternative). Let n = 1 and assume that Hypothesis 2.3.7 holds. Then
system (Sλ) is either in the limit circle case for all λ ∈ C, or in the limit point case for all λ ∈ CKR.
Although in this section we have conducted a thorough analysis of the number of
linearly independent square summable solutions of system (Sλ), we gave absolutely no
information about dim N(λ) if λ ∈ R (except for the limit circle case). Fortunately, a basic
estimate of this value is derived in Theorem 6.4.16 by using the theory of linear relations
developed in Chapter 6, see also Remark 6.4.15. Moreover, there exists an intimate
connection between the Weyl solution X(λ) with λ ∈ R and the so-called principal (or
recessive) solution of system (Sλ) in the nonoscillatory case. This connection represents
one of the goals of our current research and it will be new even in the continuous case, i.e.,
for system (2.5). Similar results were established for the second order Sturm–Liouville
differential, difference, and dynamic equations in [37] and [A23].
2.5 Illustrating examples
Now we present several examples which illustrate the results of this chapter. In the whole
section we focus on the special case of system (Sλ) with n = 1 and Sk ≡ S := I2, which
can be considered as a discrete analogue of the so-called no potential case known in
the theory of Sturm–Liouville differential equations and linear Hamiltonian differential
systems. That is, we analyze the system
zk(λ) = λVk zk(λ), k ∈ [0, ∞)Z, (2.74)
with two special choices of the matrices V ∈ C([0, ∞)Z)2×2 satisfying the assumptions
in (2.1). In each example we determine the centers and the radii of the Weyl disk and
of the limiting Weyl disk, as well as we give the corresponding limit point or limit circle
classification. We note that in the limit point case we denote by X+(λ) the Weyl solution
defined as in (2.23) with M = P+(λ).
Example 2.5.1. Let λ ∈ CKR and consider system (2.74) with the constant matrices
Vk ≡ V =
( √
ab a
−b −
√
ab
)
, k ≡ =
(
b
√
ab√
ab a
)
≥ 0 for all k ∈ [0, ∞)Z, (2.75)
where a > 0 and b ≥ 0 are given real numbers. This choice of V is naturally based on the
properties required in (2.1). Note that for ≥ 0 we need only a ≥ 0, but as we will see,
the crucial Hypothesis 2.3.7 is not satisfied when a = 0. The fundamental matrix k(λ) of
system (2.74) with α = (1, 0) has in this case the form
k(λ) = (I + λV)k
=
(
1 + kλ
√
ab kλa
−kλb 1 − kλ
√
ab
)
, k ∈ [0, ∞)Z. (2.76)
– 36 –
2.5. Illustrating examples
In view of (2.22), the two solutions Zk(λ) and Zk(λ) of system (2.74) are equal to the first
and second columns of the matrix k(λ) in (2.76), respectively. The sum in (2.25) with
N = N0 is then equal to (N0 + 1)a, which shows that Hypothesis 2.3.7 is satisfied for any
N0 ∈ [0, ∞)Z and only a > 0. From (2.36) we then get
Fk(λ) = 2kb| im(λ)|, Gk(λ) = −iδ(λ) + 2k
√
ab| im(λ)|, Hk(λ) = 2ka| im(λ)|,
which yields through (2.42) that the center and the radius of the Weyl disk Dk(λ) for
k ∈ [1, ∞)Z are
Pk(λ) = −
√
b/a +
i
2ka im(λ)
and Rk(λ) =
1
√
2ka| im(λ)|
.
By taking the limit as k → ∞ we can see that the center and the radius of the limiting
Weyl disk D+(λ) are P+(λ) = −
√
b/a and R+(λ) = 0, that is, D+(λ) =
{
−
√
b/a
}
. This
shows that system (2.74) with Vk given in (2.75) is in the limit point case for any λ ∈ CKR.
The limiting behavior of the Weyl disks is demonstrated in Figure 2.1 below. The Weyl
solution X+
k
(λ) ≡
(
1, −
√
b/a
)⊤
satisfies ||X+(λ)|| = 0 and it is the only square summable
solution (up to a nonzero constant multiple). Note that ||Z(λ)|| = ∞ and ||Z(λ)|| = ∞
for b > 0, while for b = 0 we have Z(λ) = X+(λ).
re z
im z
×
×
×
×
×
Figure 2.1: The Weyl disks Dk(λ) for k ∈ {1, 2, 4, 6, 10}, their centers, and
P+(λ) = −1 from Example 2.5.1 with a = b = 2 and λ = 0.4 + 0.4i.
▲
Although the choice of a = 0 was not possible in Example 2.5.1, we show that also in
this case the system from Example 2.5.1 is in the limit points case. This is done by using
a suitable transformation.
Example 2.5.2. Let λ ∈ CKR and consider the system from Example 2.5.1 with a = 0 and
b > 0, i.e.,
zk+1(λ) = 5(λ)zk(λ), 5(λ) =
(
1 0
−λb 1
)
, =
(
b 0
0 0
)
≥ 0, k ∈ [0, ∞)Z. (2.77)
The dependence on λ in system (2.77) is special as in (2.7) and, as we discussed in
Example 2.5.1, Hypothesis 2.3.7 is not satisfied in this case. Therefore the theory developed
– 37 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
in Section 2.3 cannot be applied, neither can be applied the results in [26] and [A4]. On
the other hand, by using the transformation
yk(λ) := T−1
zk(λ), where T :=
(
−
√
c/b −1
1 0
)
is a constant symplectic matrix with c ≥ 0, we obtain another symplectic system
yk+1(λ) = 5(λ)yk(λ), 5(λ) =
(
1 + λ
√
bc λb
−λc 1 − λ
√
bc
)
, =
(
c
√
bc√
bc b
)
, k ∈ [0, ∞)Z,
where 5(λ) := T−1 5(λ)T, see [20, Lemma 6]. Now, since b > 0 and c ≥ 0, the results in
Example 2.5.1 can be used for the above transformed system. In particular, system (2.77)
is in the limit point case for every λ ∈ CKR. Depending on the choice of the constant
c ≥ 0 in the transformation matrix T, we obtain from the reversed transformation the two
linearly independent solutions Zk(λ) =
(
−
√
c/b, 1 + kλ
√
bc
)⊤
and Zk(λ) = (−1, kλb)⊤
of system (2.77). For these solutions we easily calculate that ||Z(λ)|| = 0 when c = 0,
||Z(λ)|| = ∞ when c > 0, and ||Z(λ)|| = ∞. The corresponding Weyl solution is
X+
k
(λ) ≡ (0, 1)⊤, which obviously satisfies ||X+(λ)|| = 0. ▲
Finally, we present a system of the form as in (2.74) with nonconstant Vk, which can
be either in the limit point case or in the limit circle case.
Example 2.5.3. Let λ ∈ CKR and v ∈ C([0, ∞)Z) be a given sequence such that v0 = 0,
vk ≥ 0 for all k ∈ [0, ∞)Z, and vℓ > 0 for some index ℓ ∈ [1, ∞)Z. Define the matrix
Vk =
(
0 vk
0 0
)
for all k ∈ [0, ∞)Z and consider the system
zk+1(λ) = 5k(λ)zk(λ), 5k(λ) =
(
1 λ vk
0 1
)
, k =
(
0 0
0 vk
)
≥ 0, k ∈ [0, ∞)Z. (2.78)
The fundamental matrix of system (2.78) with α = (1, 0) is equal to
k(λ) =
(
1 λvk
0 1
)
, i.e., Zk(λ) ≡
(
1
0
)
, Zk(λ) =
(
λvk
1
)
, k ∈ [0, ∞)Z. (2.79)
This implies that Hypothesis 2.3.7 is satisfied for any N0 ∈ [ℓ − 1, ∞)Z, since the sum
in (2.25) with N = ℓ − 1 is equal to vℓ > 0. From (2.36) we get Fk(λ) = 0, Gk(λ) = −iδ(λ),
and Hk(λ) = 2vk | im(λ)|. The assumptions imply that Hk(λ) > 0 for all k ∈ [ℓ, ∞)Z, so that
the center and the radius of the Weyl disk Dk(λ) are well defined and equal to
Pk(λ) =
i
2vk im(λ)
and Rk(λ) =
1
√
2vk | im(λ)|
for all k ∈ [ℓ, ∞)Z.
If we put v∞ := limk→∞ vk = supk∈[0,∞)Z
{vk}, then v∞ > 0 and the center and the radius of the
limiting Weyl disk D+(λ) are equal to P+(λ) = i/[2v∞ im(λ)] and R+(λ) = 1/
√
2v∞ | im(λ)|.
From this one can easily conclude that system (2.78) is in the limit point case if and only
if v∞ = ∞, while it is in the limit circle case if and only if v∞ < ∞. In the latter case,
the linearly independent solutions Z(λ) and Z(λ) are square summable with ||Z(λ)|| = 0
and ||Z(λ)|| =
√
v∞. In addition, the Weyl solution X(λ) defined by (2.23) through the
fundamental matrix (λ) from (2.79) and M = m ∈ C is also square summable with the
corresponding semi-norm ||X(λ)|| =
√
v∞ |m| < ∞. The behavior of the Weyl disks is
demonstrated in Figure 2.2 below.
– 38 –
2.6. Bibliographical notes
re z
im z
×
×
×
Figure 2.2: The Weyl disks Dk(λ) for k ∈ {1, 2, 3}, their centers, and the
disk D+(λ) with P+(λ) = 5i/2 and R+(λ) =
√
5/2 from Example 2.5.3
with vk = 1 − 2−k and λ = 0.4 + 0.4i.
▲
The results of the previous example are summarized in the following statement.
Corollary 2.5.4. Let n = 1, λ ∈ CKR, v ∈ C([0, ∞)Z) be such that v0 = 0, vk ≥ 0 for
all k ∈ [0, ∞)Z, vℓ > 0 for some index ℓ ∈ [1, ∞)Z, and define v∞ := limk→∞ vk. If we put
Vk :=
(
0 vk
0 0
)
for all k ∈ [0, ∞)Z, then the following holds.
(i) System (2.74) is in the limit point case if and only if v∞ = ∞. In this case, P+(λ) = 0,
R+(λ) = 0, and the Weyl solution X+
k
(λ) ≡ (1, 0)⊤ is the only (up to a nonzero constant
multiple) square summable solution of system (2.74).
(ii) System (2.74) is in the limit circle case if and only if v∞ < ∞. In this case we have
P+(λ) = i/[2v∞ im(λ)] and R+(λ) = 1/
√
2v∞ | im(λ)|. The solutions Zk(λ) ≡ (1, 0)⊤ and
Zk(λ) = (λvk, 1)⊤ are linearly independent with ||Z(λ)|| = 0 and ||Z(λ)|| =
√
v∞.
We note that one direction in part (ii) of Corollary 2.5.4 also follows from Corollary
2.4.19, because in this case we have
∑∞
k=0 ||Sk − I||1 = 0 and
∑∞
k=0 || k ||1 = v∞ < ∞.
2.6 Bibliographical notes
The results of this chapter (including the direct proofs of Theorem 2.4.17 and Corollary
2.4.19) were published in [A15]. Moreover, their generalization to symplectic systems
on time scales was established in [A16]. In our future research we aim to construct
a particular example of system (Sλ) mentioned in Remarks 2.4.4(i) and 2.4.21, i.e., such
that dim N(λ) dim N(¯λ).
– 39 –
Chapter 2. Weyl–Titchmarsh theory for general linear dependence on spectral parameter
– 40 –
Chapter 3
Jointly varying endpoints
Perhaps the most surprising thing about mathematics is that it is so
surprising. The rules which we make up at the beginning seem ordinary
and inevitable, but it is impossible to foresee their consequences.
These have only been found out by long study, extending over many
centuries. Much of our knowledge is due to a comparatively few great
mathematicians such as Newton, Euler, Gauss, Cauchy, or Riemann;
few careers can have been more satisfying than theirs. They have contributed
something to human thought even more lasting than great
literature, since it is independent of language.
Edward Charles Titchmarsh, see [1, pg. 12]
In the previous chapter we started with the eigenvalue problem given in (2.24), which
includes the separated boundary conditions. Then, by using the fundamental matrix (λ)
determined in (2.21), we developed the theory of Weyl disks and square summable solutions
for system (Sλ). These results were achieved under the weak Atkinson condition, see
Hypothesis 2.3.7, instead of the traditional strong Atkinson condition, see Hypothesis 2.4.11
and [A4], [26,142,154]. This is crucial and absolutely essential in the context of the results
established in this chapter, where we will extend the results of Chapter 2 to problems
with general jointly varying endpoints
γ
(
z0
zN+1
)
= 0, γ ∈ Γ :=
{
γ ∈ C2n×4n
| γγ∗
= I, γ
(
−J 0
0 J
)
γ∗
= 0
}
. (3.1)
The boundary conditions in (3.1) include, among others, the periodic endpoints z0 = zN+1
or the antiperiodic endpoints z0 = −zN+1, which could not be treated by the previous
case in (2.24), see the discussion following (2.24). The method we use is based on the
augmentation of system (Sλ) into double dimension, which leads to a problem with
separated endpoints having the original boundary conditions from (3.1) as one of its
constraints. This technique is known in the literature in principle (cf. [17, 87, 88, 92, 112,
153]), but the transformation to separated endpoints presented in this chapter is much
simpler. At the same time, the transformed symplectic system no longer satisfies the
corresponding strong Atkinson condition, but only its weak form. Thus, the derivation
of the Weyl–Titchmarsh theory in Chapter 2 under the weak Atkinson condition is truly
crucial for its further extension to jointly varying endpoints.
For this general situation, we give a characterization of eigenvalues of the eigenvalue
problem determined by system (Sλ) together with the boundary conditions in (3.1), we
construct the Weyl disks, their centers and matrix radii, and also focus on properties
of square summable solutions. More precisely, we give an exact connection between
the limit point or limit circle classification of the original system (in dimension 2n) and
the augmented system (in dimension 4n). This connection reveals an interesting fact,
namely that the limiting matrix radius of the augmented system has its rank at least
– 41 –
Chapter 3. Jointly varying endpoints
n (see Theorem 3.1.11), and so it is never zero in the limit point case as one would
expect from Theorem 2.4.3. The results of this chapter (see Theorem 3.1.2) also imply the
existence of multiple eigenvalues for scalar symplectic eigenvalue problems with jointly
varying endpoints. This is known e.g. for the second order discrete Sturm–Liouville
problems with periodic endpoints in [105, Example 7.6] or [170, Theorem 2.2] and here we
extend it to discrete symplectic systems. The transformation of jointly varying endpoints
into separated endpoints will also find applications in the continuous time problems or
time scales problems, see e.g. [153]. Finally, we remark that the results shown in this
chapter were established as new even for special discrete symplectic systems, such as
those with (2.7), the Jacobi equations and symmetric three term recurrence relations, i.e.,
equations (2.9) and (2.10), and also for linear Hamiltonian difference systems.
3.1 Weyl–Titchmarsh theory for jointly varying endpoints
Throughout this chapter we use the same notation as in Chapter 2, see Notation 2.1.2.
Moreover, we emphasize the augmentation by the bold notation. For a given γ ∈ Γ and
N ∈ [0, ∞)Z we consider the eigenvalue problem
(Sλ), k ∈ [0, N]Z, λ ∈ C, (3.1). (3.2)
The eigenvalues of (3.2) are defined as for (2.24). That is, a number λ ∈ C is an eigenvalue
of problem (3.2) if, for this particular value λ, system (Sλ) has a nontrivial solution
z(λ) ∈ C([0, N + 1]Z)2n satisfying the boundary conditions in (3.1). In this case, z(λ)
is called an eigenfunction for λ and the dimension of such eigenfunctions for λ is its
geometric multiplicity. As one of the main assumptions we suppose that system (Sλ)
satisfies the strong Atkinson condition on a finite or infinite interval, see Hypothesis 3.1.1
below and Hypothesis 2.4.11, respectively.
Hypothesis 3.1.1 (Strong Atkinson condition – finite). The inequality in (2.25) is satisfied
for every nontrivial solution z(λ) ∈ C([0, N + 1]Z)2n of system (Sλ) on the discrete interval
[0, N]Z and every λ ∈ CKR.
The results of this chapter will be formulated with the aid of a particular fundamental
matrix Φ(λ) of system (Sλ) starting with the initial value Φ0(λ) = −J, which corresponds
to the fundamental matrix (λ) specified in (2.21) with the choice α = (0, I), i.e.,
Φk+1(λ) = (Sk + λVk)Φk(λ), k ∈ [0, ∞)Z, Φ0(λ) = −J, λ ∈ C. (3.3)
Our first result describes the orthogonality of the eigenfunctions and the multiplicity of
the eigenvalues of problem (3.2). It generalizes Theorem 2.2.3 to jointly varying endpoints,
compare also with [19, Theorem 2.2]. The proofs mostly follow by direct calculations in
a similar way as the corresponding results in Chapter 2. For completeness and comparison
we provide alternative proofs based on the transformation in Section 3.3.
Theorem 3.1.2. Let γ ∈ Γ be given. Then the following statements hold.
(i) A number λ ∈ C is an eigenvalue of problem (3.2) if and only if the matrix
L(λ) := γ
(
−J
ΦN+1(λ)
)
(3.4)
is singular. In this case, the eigenfunctions corresponding to the eigenvalue λ have the
form z(λ) = Φ(λ)d on [0, N + 1]Z with a nonzero d ∈ Ker L(λ). Moreover, the geometric
multiplicity of λ is equal to its algebraic multiplicity, i.e., to the value of dim Ker L(λ).
– 42 –
3.1. Weyl–Titchmarsh theory for jointly varying endpoints
(ii) Under Hypothesis 3.1.1, the eigenvalues of problem (3.2) are real and the eigenfunctions
corresponding to different eigenvalues are orthogonal with respect to the semi-inner product
⟨·, ·⟩ ,N defined in (2.26).
Proof. The statement follows from Theorem 3.3.5 with (3.29) and Corollary 3.3.3. ■
By Theorem 3.1.2, the multiplicities of the eigenvalues of problem (3.2) are at most 2n,
compared to the separated endpoints case in Theorem 2.2.3 in which the multiplicities
of the eigenvalues are at most n. It implies that in the scalar case (i.e., for n = 1) there
may exist multiple eigenvalues of problem (3.2). This phenomenon was observed in [105,
Example 7.6] and later justified in [170, Theorem 2.2] for the periodic discrete Sturm–
Liouville eigenvalue problem, see also Example 3.2.4.
Remark 3.1.3. When system (Sλ) has the special structure shown in (2.7), it can be deduced
from [152, Corollary 4.6] that the total number of eigenvalues of (3.2) is equal to the
dimension of the space of admissible functions for the associated discrete quadratic
functional. In some even more special cases, such as for the second order Sturm–Liouville
difference equations with periodic or antiperiodic endpoints, this exact number of the
eigenvalues of problem (3.2) is derived in [170, Theorem 4.2] or [145, Theorem 4.1]. The
result in [152, Corollary 4.6] is based on the Rayleigh principle for system (Sλ) with the
boundary condition from (3.1), compare also with [153, Theorem 3.2], and on the fact that
the space of admissible functions is independent of λ, which follows from the special
structure in (2.7). As the Rayleigh principle for eigenvalue problems (3.2) is not known
and the space of admissible functions is in this case not constant in λ, the question about
the total number of eigenvalues of problem (3.2) remains open for the general linear
dependence on λ. On the other hand, the oscillation theorem for discrete symplectic
eigenvalue problems with jointly varying endpoints in [148, Theorem 6.13] yields that the
total number of the eigenvalues of problem (3.2) is less or equal to (N + 3)n.
Next we define the Weyl–Titchmarsh M(λ)-function for problem (3.2), compare with
identity (2.27). For k ∈ [0, N + 1]Z and λ ∈ C we set
Mk(λ) := −
[
γ
(
−J
Φk(λ)
) ]−1
γ
(
J
Φk(λ)
)
J, (3.5)
whenever the inverse above exists. In particular, we can see from Theorem 3.1.2 that
Mk(λ) is well defined for every λ ∈ CKR and k = N +1, when Hypothesis 3.1.1 holds. The
following statement generalizes Lemma 2.2.5 to jointly varying endpoints.
Theorem 3.1.4. Let γ ∈ Γ, λ ∈ C, and k ∈ [0, ∞)Z. If Mk(λ) and Mk(¯λ) exist, then we have
M∗
k
(λ) = Mk(¯λ). Moreover, Mk(·) is an analytic function in its argument λ.
Proof. This result follows from (3.36) and (3.33) via Lemma 2.2.5. ■
For any M ∈ C2n×2n we define the Weyl-solution X(λ) of (Sλ) with values in C4n×2n by
Xk(λ) :=
1
√
2
(
J −J
Φk(λ) Φk(λ)
) (
J
M
)
. (3.6)
It then follows, compare with Remark 2.2.6(i), that γXk(λ) = 0 if and only if the matrix M
equals to Mk(λ) defined in (3.5).
– 43 –
Chapter 3. Jointly varying endpoints
One of the central concepts of this chapter is the E(M)-function with values in C2n×2n,
through which we later on define the Weyl disks. For M ∈ C2n×2n we put
Ek(M) := iδ(λ)X∗
k(λ)
(
−J 0
0 J
)
Xk(λ) =
(
J
M
)∗ (
Hk(λ) Gk(λ)
Gk(λ) Hk(λ)
) (
J
M
)
, (3.7)
Hk(λ) := 1
2 iδ(λ)[Φ∗
k(λ)JΦk(λ) − J], Gk(λ) := Hk(λ) + iδ(λ)J. (3.8)
From (iJ)∗ = iJ one can easily see that Ek(M), Hk(λ), and Gk(λ) are Hermitian matrices.
Moreover, we have H0(λ) = 0 and the Lagrange identity in Theorem 2.1.7 implies the
following crucial equalities
Ek(M) = −2δ(λ) im(M) + | im(λ)|(M∗
− J)


k−1∑
j=0
Φ∗
j+1(λ) j Φj+1(λ)


(M + J), (3.9)
Hk(λ) = | im(λ)|
k−1∑
j=0
Φ∗
j+1(λ) j Φj+1(λ), (3.10)
compare with (2.32), (2.36), (2.33), and (2.37), respectively. Since Φ(λ) represents a fundamental
matrix of system (Sλ), equality (3.10) justifies the following result.
Theorem 3.1.5. If Hypothesis 2.4.11 holds, then the matrix Hk(λ) is positive definite for every
λ ∈ CKR and k ∈ [N3 + 1, ∞)Z. In addition, for such k we have (suppressing the argument λ)
Ek(M) = −J
(
Hk − Gk H−1
k Gk
)
J +
(
M∗
− JGk H−1
k
)
Hk
(
M + H−1
k Gk J
)
. (3.11)
Proof. The invertibility of Hk(λ) for all k ∈ [N3 + 1, ∞)Z follows from (3.10) and Hypothesis
2.4.11. Moreover, identity (3.11) is a consequence of (3.37) and (3.39). ■
For any λ ∈ CKR we now define the Weyl disk Dk(λ) and the Weyl circle Ck(λ) as
Dk(λ) :=
{
M ∈ C2n×2n
| Ek(M) ≤ 0
}
and Ck(λ) :=
{
M ∈ C2n×2n
| Ek(M) = 0
}
,
compare with Definition 2.3.1. The following result provides some properties of the
elements in Dk(λ) and Ck(λ). It is a generalization of Theorems 2.3.2 and 2.3.3 to jointly
varying endpoints.
Theorem 3.1.6. Let λ ∈ CKR, k ∈ [0, ∞)Z, and M ∈ C2n×2n. Then the following hold.
(i) The matrix M ∈ Ck(λ) if and only if there exists γ ∈ Γ such that γXk(λ) = 0. In this case,
we have with such a matrix γ that M = Mk(λ), whenever the matrix Mk(λ) exists.
(ii) The matrix M satisfies Ek(M) < 0 if and only if there exists γ ∈ C2n×4n such that
iδ(λ)γ
(
−J 0
0 J
)
γ∗ > 0 and γXk(λ) = 0. In this case, we have with such a matrix γ
that M = Mk(λ), whenever the matrix Mk(λ) exists, and γ can be chosen so that γγ∗ = I.
(iii) We have Ek(−J) = −2δ(λ)iJ, i.e., −J Dk(λ).
Proof. Statements (i) and (ii) follow by Theorems 2.3.2 and 2.3.3 from the facts that the
sets Dk(λ) and Ck(λ) coincide respectively with the Weyl disk and Weyl circle in (3.38).
Statement (iii) is verified by direct calculation from (3.7), because the matrix iJ is indefinite.
■
– 44 –
3.1. Weyl–Titchmarsh theory for jointly varying endpoints
The center Pk(λ) and the matrix radius Rk(λ) of the Weyl disk Dk(λ) are defined as the
2n × 2n matrices
Pk(λ) := −H−1
k (λ)Gk(λ)J = −J + iδ(λ)H−1
k (λ) and Rk(λ) := H−1/2
k
(λ), (3.12)
whenever Hk(λ) is invertible, i.e., whenever Hk(λ) > 0, compare with (2.42). Note that
from (3.8) and (1.5) we get the series expansion
Pk(λ) = −J + 2J
∞∑
j=0
[
− Φ∗
k(λ)JΦk(λ)J
]j
, when sprad Φ∗
k(λ)JΦk(λ)J < 1.
The following theorem provides the most important geometric properties of the Weyl
disks, including their nested property, closedness, and convexity. It is a generalization of
Theorems 2.3.6 and 2.3.8 to the case of jointly varying endpoints. Hence we denote by 777
and 888 the sets of all unitary and contractive 2n × 2n complex matrices, respectively, i.e.,
777 := {U ∈ C2n×2n
| U∗
U = I} and 888 := {V ∈ C2n×2n
| V∗
V ≤ I}. (3.13)
Theorem 3.1.7. Let λ ∈ CKR. Then Dk(λ) ⊆ Dj(λ) for every k, j ∈ [0, ∞)Z with k ≥ j. In
addition, under Hypothesis 2.4.11 we have for every k ∈ [N3 + 1, ∞)Z the representations
Dk(λ) =
{
Pk(λ) + Rk(λ)VRk(¯λ) | V ∈ 888
}
,
Ck(λ) =
{
Pk(λ) + Rk(λ)URk(¯λ) | U ∈ 777
}
.
Consequently, the Weyl disks Dk(λ) are closed and convex for every k ∈ [N3 + 1, ∞)Z.
Proof. The result follows from (3.40), (3.41), and (3.42) combined with Corollary 3.3.4. ■
The latter theorem implies that the intersection of all Weyl disks Dk(λ) over the discrete
interval [N3 + 1, ∞)Z is a nonempty, closed, and convex set. This yields that the limiting
Weyl disk has the form
D+(λ) :=
∩
k∈[N3+1,∞)Z
Dk(λ) =
{
P+(λ) + R+(λ)VR+(¯λ) | V ∈ 888
}
,
where P+(λ) and R+(λ) are the 2n × 2n matrices defined by
P+(λ) := lim
k→∞
Pk(λ), R+(λ) := lim
k→∞
Rk(λ) ≥ 0. (3.14)
They are called the center and the matrix radius of the limiting Weyl disk D+(λ), compare
with Definition 2.3.10. Note that the convergence of Pk(λ) and Rk(λ) can be seen from
their definitions given in (3.12) and equality (3.10).
Remark 3.1.8. If Hinv
+ (λ) denotes the limit of H−1
k
(λ) as k → ∞, which exists by (3.10), then
the formulas in (3.14) for the center and matrix radius of the limiting Weyl disk reduce to
P+(λ) = −J + iδ(λ)Hinv
+ (λ), R+(λ) =
[
Hinv
+ (λ)
]1/2
. (3.15)
The next result is a generalization of Corollary 2.3.12 to jointly varying endpoints.
Note that as in Theorem 3.1.6(iii) we have −J D+(λ).
– 45 –
Chapter 3. Jointly varying endpoints
Theorem 3.1.9. Let λ ∈ CKR, M ∈ C2n×2n, and suppose that Hypothesis 2.4.11 holds. Then M
belongs to the limiting Weyl disk D+(λ) if and only if
(M∗
− J)


∞∑
k=0
Φ∗
k+1(λ) k Φk+1(λ)

 (M + J) ≤
2 im(M)
im(λ)
.
Proof. This statement follows from (3.9), or alternatively by Corollary 2.3.12 from (3.43)
and the definition of the Weyl solution in (3.36) and (3.33). ■
Remark 3.1.10. The limiting Weyl circle C+(λ) can be introduced as the boundary of the
limiting Weyl disk D+(λ). Then M ∈ C+(λ) if and only if any of the following two
equivalent conditions hold, compare with Remark 2.3.17(ii),
(M∗
− J)


∞∑
k=0
Φ∗
k+1(λ) k Φk+1(λ)

 (M + J) =
2 im(M)
im(λ)
,
lim
k→∞
X∗
k(λ)
(
−J 0
0 J
)
Xk(λ) = 0.
Finally, let us discuss some properties of square summable solutions of system (Sλ).
The following result is quite surprising in the sense that one would expect to have
R+(λ) = 0 in the limit point case, see Theorem 2.4.3. To the contrary, due to the augmented
structure of the matrix R+(λ), which has dimension 2n, it is the rank of R+(λ) alone which
determines the number of linearly independent square summable solutions of system (Sλ),
compare with Theorem 2.4.8. In the result below we show that rank R+(λ) ≥ n, so that the
equality rank R+(λ) = n must necessarily hold in the limit point case. This fact is stated
in Corollary 3.1.12 below and also illustrated in Example 3.2.6.
Theorem 3.1.11. Let λ ∈ CKR and suppose that Hypothesis 2.4.11 holds. Then system (Sλ) has
exactly rank R+(λ) linearly independent square summable solutions, i.e.,
n ≤ dim N(λ) = rank R+(λ) ≤ 2n. (3.16)
Proof. The statement in (3.16) is proven in Theorem 3.3.6 and (3.44). ■
The meaning of Theorem 3.1.11 can be explained also directly from (3.10) and Remark
3.1.8. In particular, by (3.15), the rank of R+(λ) is equal to the number of positive
eigenvalues of the matrix Hinv
+ (λ) from Remark 3.1.8 and this number is the same as
the number of the eigenvalues of Hk(λ), which tend to a finite limit as k → ∞. Consequently,
equality (3.10) shows that it is equal to the number of linearly independent
square summable solutions of system (Sλ).
Corollary 3.1.12. Let λ ∈ CKR and suppose that Hypothesis 2.4.11 holds. Then system (Sλ) is
in the limit point case if and only if rank R+(λ) = n, while (Sλ) is in the limit circle case if and
only if rank R+(λ) = 2n.
3.2 Examples
Now we examine several examples which illustrate the theory presented in the previous
section. In particular, we consider the periodic and antiperiodic boundary conditions as
in [153, Remark 6.17] and the corresponding M(λ)-function.
– 46 –
3.2. Examples
Example 3.2.1. For the periodic endpoints z0 = zN+1 we take γ = 1√
2
(J, −J) ∈ Γ. In this
case, the matrix in (3.4) is
L(λ) = 1√
2
J[ΦN+1(λ) + J]
Then, by Theorem 3.1.2, a number λ ∈ C is an eigenvalue of problem (3.2) if and only if
the matrix ΦN+1(λ)+J is singular, and the number dim Ker[ΦN+1(λ)+J] is its multiplicity.
Moreover, the M(λ)-function in (3.5) reduces to
M
[p]
k
(λ) = −[Φk(λ) + J]−1
[Φk(λ) − J]J. (3.17)
▲
Example 3.2.2. For the antiperiodic endpoints z0 = −zN+1 we take γ = 1√
2
(J, J) ∈ Γ. In
this case we have
L(λ) = 1√
2
J[ΦN+1(λ) − J]
and, by Theorem 3.1.2, a number λ ∈ C is an eigenvalue of problem (3.2) if and only if
the matrix ΦN+1(λ) − J is singular. The multiplicity of λ is then dim Ker[ΦN+1(λ) − J]. In
addition, the M(λ)-function in (3.5) now has the form
M
[ap]
k
(λ) = −[Φk(λ) − J]−1
[Φk(λ) + J]J. (3.18)
▲
The M(λ)-functions M
[p]
k
(λ) and M
[ap]
k
(λ) from (3.17) and (3.18) for the periodic and
antiperiodic endpoints are closely related, as we show in the next interesting statement.
Let p be the set of all eigenvalues of the periodic problem (3.2) with γ = γp from
Example 3.2.1. Similarly, let ap be the set of all eigenvalues of the antiperiodic problem
(3.2) with γ = γap from Example 3.2.2. Then by Theorem 3.1.2(ii) we have p ∪ ap ⊆ R
under Hypothesis 3.1.1.
Corollary 3.2.3. Let k ∈ [0, N + 1]Z and λ ∈ C be fixed. The matrices M
[p]
k
(λ) and M
[ap]
k
(λ) given
in (3.17) and (3.18) satisfy the following conditions.
(i) If Φk(λ) + J is invertible, then rank M
[p]
k
(λ) = rank[Φk(λ) − J]. In particular, this equality
holds at k = N + 1 for every λ p.
(ii) If Φk(λ)−J is invertible, then rank M
[ap]
k
(λ) = rank[Φk(λ)+J]. In particular, this equality
holds at k = N + 1 for every λ ap.
(iii) If Φk(λ) + J and Φk(λ) − J are invertible, then M
[p]
k
(λ) and M
[ap]
k
(λ) are also invertible and
satisfy the equalities
[
M
[p]
k
(λ)J
]−1
= M
[ap]
k
(λ)J and det M
[p]
k
(λ) × det M
[ap]
k
(λ) = 1.
In particular, these equalities hold at k = N + 1 for every λ p ∪ ap.
As an addendum to Corollary 3.2.3 we derive the series representations for M
[p]
k
(λ) and
M
[ap]
k
(λ). If we expand the inverses in (3.17) and (3.18) by (1.5), then under the condition
sprad Φk(λ)J < 1 we obtain
M
[p]
k
(λ) = 2J
( ∞∑
j=0
[
Φk(λ)J
]j
)
− J and M
[ap]
k
(λ) = 2J
( ∞∑
j=0
[
− Φk(λ)J
]j
)
− J.
– 47 –
Chapter 3. Jointly varying endpoints
Now, we illustrate our results on the scalar symplectic system
zk+1(λ) =
(
1 1
−λ 1 − λ
)
zk(λ) with k ≡ :=
(
1 0
0 0
)
, (3.19)
i.e., Sk ≡
(
1 1
0 1
)
and Vk ≡
(
0 0
−1 −1
)
. This system corresponds to the second order Sturm–
Liouville difference equation (2.8) with m = n = 1, P[1]
k
= Wk ≡ 1, and P[0]
k
≡ 0, i.e.,
− (pk yk) + qk yk+1 = λwk yk+1, where pk = wk ≡ 1, qk ≡ 0. (3.20)
System (3.19) satisfies the strong Atkinson condition in Hypothesis 3.1.1 or 2.4.11 with
N3 = 1, as can be easily verified.
Example 3.2.4. Let us consider the scalar eigenvalue problem with periodic endpoints
(3.19), k ∈ [0, 3]Z, z0 = z4, (3.21)
i.e., we look for the solutions of system (3.19) with period 4. This problem corresponds
to the periodic Sturm–Liouville eigenvalue problem (3.20) with k ∈ [0, 3]Z and y0 = y4,
y0 = y4, which was studied in [105, Example 7.6]. It was shown in the latter reference
that λ = 2 is a double eigenvalue of problem (3.21) by finding two linearly independent
eigenfunctions. The results in Theorem 3.1.2 and Example 3.2.1 confirm this conclusion.
The fundamental matrix Φ(λ) from (3.3) now satisfies
Φ2(λ) =
(
−λ + 2 λ − 1
λ2 − 3λ + 1 −λ2 + 2λ
)
, (3.22)
Φ4(λ) =
(
−λ3 + 6λ2 − 10λ + 4 λ3 − 5λ2 + 6λ − 1
λ4 − 7λ3 + 15λ2 − 10λ + 1 −λ4 + 6λ3 − 10λ2 + 4λ
)
. (3.23)
This yields that det[Φ4(λ) + J] = −λ(λ − 4)(λ − 2)2. Thus, by Theorem 3.1.2 and Example
3.2.1, λ = 2 is indeed a double eigenvalue of problem (3.21), and the columns of Φ(2)
are the two linearly independent eigenfunctions. Note that it holds Φ4(2) = −J = Φ0(2).
The other eigenvalues of problem (3.21) are λ = 0 with the eigenfunction Φ(0)(0, 1)⊤ and
λ = 4 with the eigenfunction Φ(4)(2, 1)⊤. ▲
Example 3.2.5. Let us consider again system (3.19), but now only on the interval [0, 2]Z
and with the antiperiodic boundary conditions z0 = −z2. From equality (3.22) we see that
det[Φ2(λ) − J] = (λ − 2)2. Hence, by Theorem 3.1.2 and Example 3.2.2, λ = 2 is a double
eigenvalue of this problem with the columns of Φ(2) as the two linearly independent
eigenfunctions. Note that it holds Φ2(2) = J = −Φ0(2). This problem then does not have
any other eigenvalues. ▲
In the last example we calculate the rank of the limiting radius R+(λ) and compare it
with the corresponding number of linearly independent square summable solutions.
Example 3.2.6. We examine system (3.19) on the discrete interval [0, ∞)Z with a particular
choice of λ0 ∈ CKR. We show that system (3.19) with λ = λ0 is in the limit point case, so
that by Corollary 2.4.22 it is in the limit point case for every λ ∈ CKR. Let λ0 = 2 + 2i
√
3,
i.e., system (3.19) reduces to the second order difference equation yk+2 +2i
√
3 yk+1 + yk = 0
on [0, ∞)Z. The roots of the corresponding characteristic polynomial are ν± := (±2 −
√
3)i,
so that the fundamental matrix Φ(λ) of system (3.19) satisfying (3.3) has the form
Φk(λ0) =
(
νk
+ νk
−
νk
+(ν+ − 1) νk
−(ν− − 1)
)
T, where T := 1
4
(
−i −2 −
√
3 + i
i −2 +
√
3 − i
)
. (3.24)
– 48 –
3.3. Augmented symplectic system
By (3.10), we obtain with µ± := |ν± |2 = 7 ∓ 4
√
3 that
Hk(λ0) = 2
√
3 T∗
k−1∑
j=0
(
|ν+ |2j+2 (¯ν+ν−)j+1
(ν+ ¯ν−)j+1 |ν− |2j+2
)
T = 2
√
3 T∗
k−1∑
j=0
(
(µ+)j+1 (−1)j+1
(−1)j+1 (µ−)j+1
)
T.
Since each entry of Hk(λ0) represents a geometric series, it can be evaluated explicitly as
Hk(λ0) = 2
√
3 T∗
(
(µ+)[1 − (µ+)k]/(1 − µ+) [(−1)k − 1]/2
[(−1)k − 1]/2 (µ−)[1 − (µ−)k]/(1 − µ−)
)
T.
Therefore, the matrix Hk(λ0) is indeed invertible (and positive definite) on the discrete
interval [N3 + 1, ∞)Z = [2, ∞)Z and, by Remark 3.1.8, we have
Hinv
+ (λ0) = lim
k→∞
H−1
k (λ0) =
(
4 2 +
√
3 − i
2 +
√
3 + i 2 +
√
3
)
,
P+(λ0) = −J + iHinv
+ (λ0) =
(
4i (2 +
√
3)i
(2 +
√
3)i (2 +
√
3)i
)
.
The matrix R+(λ0) can also be calculated explicitly by (3.15), but it is not really important.
We can find the eigenvalues of R+(λ0) as the nonnegative square roots of the eigenvalues
of Hinv
+ (λ0). Namely, since the eigenvalues of the matrix Hinv
+ (λ0) are 0 and 6 +
√
3, we
obtain rank R+(λ0) = 1 and system (3.19) is in the limit point case, by Corollary 3.1.12.
The square summable solution of (3.19) is then given as the second component of the
columns of the Weyl solution X+(λ0) from (3.6) with M = P+(λ0). That is, the columns
of the matrix Φ(λ0) × Hinv
+ (λ0) are square summable. But since Hinv
+ (λ0) is singular, it
follows that the square summable solutions of (3.19) are generated by exactly one column
of the matrix Φ(λ0) × Hinv
+ (λ0). On the other hand, since |ν+ | < 1, one can identify the first
column of Φ(λ0) × T−1 in (3.24) as the square summable solution of (3.19). ▲
3.3 Augmented symplectic system
We will show that problem (3.2) is equivalent to a certain eigenvalue problem in dimension
4n with separated endpoints. At first, let us define the 4n × 4n matrices
Sk :=
(
I 0
0 Sk
)
, Vk :=
(
0 0
0 Vk
)
, k :=
(
0 0
0 k
)
≥ 0, (3.25)
k(λ) := 1√
2
(
J −J
Φk(λ) Φk(λ)
)
, J :=
(
−J 0
0 J
)
, K :=
(
0 J
J 0
)
, (3.26)
where Φ(λ) is the fundamental matrix of system (Sλ) specified in (3.3). Then one can
easily verify that J∗ = −J = J−1, K∗ = −K = K−1, and
S∗
k JSk = J, S∗
k JVk is Hermitian, V∗
k JVk = 0, k = JVk JS∗
k J.
With this setting, we introduce the augmented symplectic system
zk+1(λ) = (Sk + λVk)zk(λ). (Sλ)
It follows that (λ) is a fundamental matrix of system (Sλ), because
k(λ) =
(
I 0
0 Φk(λ)J
)
Q with Q := 0(λ) = 1√
2
(
J −J
−J −J
)
and det Q = 1,
– 49 –
Chapter 3. Jointly varying endpoints
Moreover, by analogy with Lemma 2.1.6 we obtain on [0, ∞)Z the identities
∗
k(λ)J k(¯λ) = K, k(¯λ)K ∗
k(λ) = J, and ∗
0(λ) 0(λ) = I.
Note that the latter equality means that the matrix 0(λ) is unitary. The double size of
system (Sλ) implies that we consider all vector solutions z(λ) of system (Sλ) in dimension
4n and matrix-valued solutions Z(λ) of system (Sλ) in dimension 4n × 2n. It then follows
that they have the form
zk(λ) =
(
d
zk(λ)
)
and Zk(λ) =
(
D E
Z[1]
k
(λ) Z[2]
k
(λ)
)
, (3.27)
where d ∈ Cn and D, E ∈ C2n×n are constant, while the sequences z(λ) ∈ C([0, ∞)Z)2n and
Z[1]
(λ), Z[2]
(λ) ∈ C([0, ∞)Z)2n×n solve system (Sλ). It turns out that the main properties of
system (Sλ) and its solutions, such as those in Section 2.1 or [A4, Section 2], are preserved
for the augmented system (Sλ). In particular, the coefficients of system (Sλ) satisfy the
following identities
(Sk + λVk)∗
J(Sk + ¯λVk) = J, (Sk + λVk)−1
= −J(S∗
k + λV∗
k)J,
and we have also the Lagrange identity for two 4n × m matrix solutions of systems (Sλ)
and (Sν), i.e., for all λ, ν ∈ C and k ∈ [0, ∞)Z it holds
Z∗
k+1(λ)JZk+1(ν) = Z∗
0(λ)JZ0(ν) + (¯λ − ν)
k∑
j=0
Z∗
j+1(λ) j Zj+1(ν). (3.28)
Given γ ∈ from (3.1), we define the 2n × 4n matrices α, β ∈ := Γ by
α := 1√
2
(
−I I
)
and β := γ. (3.29)
Since vector solutions of system (Sλ) can be written as in (3.27), the choice of α in (3.29)
implies that the solutions of (Sλ) satisfying αz0(λ) = 0 have necessarily the form
zk(λ) =
(
z0(λ)
zk(λ)
)
, (3.30)
where z(λ) ∈ C([0, ∞)Z)2n solves system (Sλ). Conversely, every solution z(λ) of system
(Sλ) yields through formula (3.30) a solution z(λ) of system (Sλ) such that αz0(λ) = 0.
Therefore, the original eigenvalue problem (3.2) is equivalent to the augmented eigenvalue
problem with separated endpoints
(Sλ), k ∈ [0, N]Z, λ ∈ C, αz0 = 0, βzN+1 = 0. (3.31)
In addition, the form of k in (3.25) implies that the semi-norm of any augmented solution
z(λ) is the same as the semi-norm of the corresponding solution z(λ), because
⟨z, ˜z⟩ ,N :=
N∑
k=0
z∗
k+1 k ˜zk+1 =
N∑
k=0
z∗
k+1 k ˜zk+1 = ⟨z, ˜z⟩ ,N with zk =
(
z0
zk
)
, ˜zk =
(
˜z0
˜zk
)
. (3.32)
– 50 –
3.3. Augmented symplectic system
In order to apply the theory from Sections 2.2 and 2.3, we need to find the fundamental
matrix (λ) of the augmented system (Sλ) such that 0(λ) = (α∗, −Jα∗), see (2.21)
and (2.22). That is,
k(λ) =
(
Zk(λ), Zk(λ)
)
, Zk(λ) := 1√
2
(
−I
Φk(λ)J
)
, Zk(λ) := 1√
2
(
−J
Φk(λ)
)
. (3.33)
with Φ(λ) being the fundamental matrix of (Sλ) used in (3.26). The above transformation
then yields the results for the eigenvalue problem (3.31) in terms of (λ). When translating
these results to the data of the original problem (3.2) we use the fundamental matrix (λ)
in (3.26). Its relationship with (λ) is given by the equality
k(λ) = k(λ)L, where L :=
(
J 0
0 I
)
. (3.34)
From (3.34) we can see that the second column Z(λ) of (λ) and (λ) is the same. Finally,
the theory also requires the following weak Atkinson-type conditions for system (Sλ),
compare with Hypotheses 2.2.2 and 2.3.7.
Hypothesis 3.3.1 (Weak augmented Atkinson condition – finite). For any λ ∈ CKR every
column z(λ) of the solution Z(λ) satisfies
N∑
k=0
z∗
k+1(λ) k zk+1(λ) > 0. (3.35)
Hypothesis 3.3.2 (Weak augmented Atkinson condition – infinite). There exists a number
N3 ∈ [0, ∞)Z such that each column z(λ) of Z(λ) satisfies inequality (3.35) with N = N3
for every λ ∈ CKR.
When we write the weak Atkinson condition in (3.35) in terms of the data of the
original problem (3.2), we get by (3.25) and (3.33) that
N∑
k=0
Z
∗
k+1(λ) k Zk+1(λ) =
N∑
k=0
Φ∗
k+1(λ) k Φk+1(λ) > 0.
This shows that the conditions in Hypotheses 3.1.1 and 3.3.1 are intimately connected as
stated in the following corollary.
Corollary 3.3.3. System (Sλ) satisfies the strong Atkinson condition on the discrete interval
[0, N]Z (Hypothesis 3.1.1) if and only if the augmented system (Sλ) satisfies the corresponding
weak Atkinson condition on [0, N]Z (Hypothesis 3.3.1).
Similar connection is true also for Hypotheses 2.4.11 and 3.3.2. This fact justifies the
use of the same index N3 in both conditions.
Corollary 3.3.4. System (Sλ) satisfies the strong Atkinson condition on the discrete interval
[0, ∞)Z (Hypothesis 2.4.11) if and only if the augmented system (Sλ) satisfies the corresponding
weak Atkinson condition on [0, ∞)Z (Hypothesis 3.3.2).
In particular, we can see why assuming the weak Atkinson condition in Chapter 2 is
really essential – the transformation of the problem (3.2) with jointly varying endpoints,
which satisfies the strong Atkinson condition, leads to the augmented problem (3.31)
satisfying the corresponding weak Atkinson condition. Therefore, one can simply apply
the previous results on separated endpoints to the augmented problem and then transform
the obtained results back to the data of the original problem (3.2).
The next theorem provides basic properties of problem (3.31).
– 51 –
Chapter 3. Jointly varying endpoints
Theorem 3.3.5. Let α, β ∈ be given. Then the following statements hold.
(i) A number λ ∈ C is an eigenvalue of (3.31) if and only if the matrix L(λ) := βZN+1(λ) is
singular. In this case, the eigenfunctions of problem (3.31) corresponding to the eigenvalue
λ have the form z = Zk(λ)d on [0, N + 1]Z with a nonzero d ∈ Ker L(λ). Moreover, the
geometric and algebraic multiplicities of λ coincide and are equal to dim Ker L(λ).
(ii) Under Hypothesis 3.3.1, the eigenvalues of problem (3.31) are real and the eigenfunctions
corresponding to different eigenvalues are orthogonal with respect to the semi-inner product
⟨·, ·⟩ ,N defined in (3.32).
Proof. The statement follows from Theorem 2.2.3, when it is applied to the augmented
eigenvalue problem (3.31). ■
Following Definitions 2.2.1 and 2.2.4, we define the Weyl solution for system (Sλ)
corresponding to M ∈ C2n×2n and the M(λ)-function associated with problem (3.31) as
Xk(λ) := k(λ)
(
I
M
)
= k(λ)
(
J
M
)
and Mk(λ) := −
[
βZk(λ)
]−1
βZk(λ), (3.36)
where (λ), Z(λ), and Z(λ) are given in (3.33). In addition, for M ∈ C2n×2n we define the
E(M)-function by
Ek(M) := iδ(λ)X∗
k(λ)JXk(λ) =
(
I
M
)∗ (
Fk(λ) G∗
k
(λ)
Gk(λ) Hk(λ)
) (
I
M
)
, (3.37)
where Fk(λ), Gk(λ), and Hk(λ) are the 2n × 2n matrices
Fk(λ) := iδ(λ)Z∗
k(λ)JZk(λ) = J∗
Hk(λ)J,
Gk(λ) := iδ(λ)Z
∗
k(λ)JZk(λ) = Hk(λ)J − iδ(λ),
Hk(λ) := iδ(λ)Z
∗
k(λ)JZk(λ) = 1
2 iδ(λ)[Φ∗
k(λ)JΦk(λ) − J].
As in Definition 2.3.1, the Weyl disk Dk(λ) and the Weyl circle Ck(λ) are defined by
Dk(λ) :=
{
M ∈ C2n×2n
| Ek(M) ≤ 0
}
, Ck(λ) :=
{
M ∈ C2n×2n
| Ek(M) = 0
}
. (3.38)
Note that under Hypothesis 3.3.2 the matrices Hk(λ) are positive definite (and hence
invertible) for all k ∈ [N3 + 1, ∞)Z, because by (3.28) we have
Hk(λ) = 2| im(λ)|
k−1∑
j=0
Z
∗
j+1(λ) j Zj+1(λ). (3.39)
By Theorem 2.3.8, the Weyl disk and the Weyl circle possess the representations
Dk(λ) =
{
Pk(λ) + Rk(λ)VRk(¯λ) | V ∈ 888
}
, (3.40)
Ck(λ) =
{
Pk(λ) + Rk(λ)URk(¯λ) | U ∈ 777
}
, (3.41)
where 888 and 777 are, respectively, the sets of all complex contractive and unitary 2n × 2n
matrices introduced in (3.13) and where the center Pk(λ) and the matrix radius Rk(λ) are
defined by
Pk(λ) := −H−1
k (λ)Gk(λ) = −J + iδ(λ)H−1
k (λ), Rk(λ) := H−1/2
k
(λ). (3.42)
– 52 –
3.3. Augmented symplectic system
Therefore, the Weyl disks Dk(λ) are closed, convex, and nested, which implies that the
limiting Weyl disk
D+(λ) :=
∩
k∈[N3+1,∞)Z
Dk(λ) =
{
P+(λ) + R+(λ)VR+(¯λ) | V ∈ 888
}
(3.43)
exists and is nonempty, closed, and convex as well. By using Theorem 2.3.9 and the
monotonicity of Hk(λ) shown in (3.39), the center and the matrix radius of D+(λ) are
P+(λ) := lim
k→∞
Pk(λ) and R+(λ) := lim
k→∞
Rk(λ) ≥ 0.
The final results of this section are devoted to the square integrable solutions of
the augmented system (Sλ). Let ℓ2
be the space of all square summable sequences
z ∈ C([0, ∞)Z)4n with the corresponding semi-norm defined as
||z|| < ∞, where ||z|| :=
( ∞∑
k=0
z∗
k+1 k zk+1
)1/2
= lim
N→∞
√
⟨z, z⟩ ,N.
For every λ ∈ C we denote the space of all square summable solutions of system (Sλ) by
N(λ) :=
{
z ∈ ℓ2
| z solves (Sλ)
}
.
If Hypothesis 3.3.2 is satisfied, we know from Theorem 2.4.1 that the dimension of N(λ)
is at least 2n, or more precisely
dim N(λ) = 2n + rank R+(λ), (3.44)
by Theorem 2.4.8. On the other hand, the analysis of the structure of the square summable
solutions of the augmented system (Sλ) yields the following result.
Theorem 3.3.6. Let λ ∈ CKR and suppose that Hypothesis 3.3.2 holds. Then
3n ≤ 2n + dim N(λ) = dim N(λ) ≤ 4n (3.45)
n ≤ dim N(λ) = n + rank R+(λ) = rank R+(λ) ≤ 2n. (3.46)
Proof. Let ej ∈ C2n be the j-th canonical unit vector for j ∈ {1, . . . , 2n}. Then system (Sλ)
possesses constant solutions z[j]
:= (e∗
j
, 0∗)∗ ∈ C4n for all j ∈ {1, . . . , 2n}, which certainly
belong to N(λ), because they satisfy ||z[j]
|| = 0. In addition, any square summable
solution z ∈ N(λ) naturally generates a square summable solution z = (0∗, z∗)∗ ∈ N(λ),
which is linearly independent with the above defined solutions z[1]
, . . . , z[2n]
. This yields
that dim N(λ) = 2n+dim N(λ). Hence identity (3.45) follows from Corollary 3.3.4 and the
inequality dim N(λ) ≥ n in Theorem 2.4.1. Identity (3.46) is then only a direct consequence
of equality (3.44). ■
Combining Theorem 3.3.6 and Corollaries 3.3.4 and 2.4.20 then yields the next result.
Corollary 3.3.7. Let λ0 ∈ CKR and suppose that Hypothesis 2.4.11 is satisfied. Then we have
dim N(λ0) = 3n if and only if system (Sλ0
) is in the limit point case. Similarly, dim N(λ0) = 4n
if and only if system (Sλ0
) is in the limit circle case. Moreover, if it holds dim N(λ1) = 4n for
some λ1 ∈ C, then dim N(λ) = 4n for all λ ∈ C.
– 53 –
Chapter 3. Jointly varying endpoints
We can now see that the rank of the limiting matrix radius R+(λ) can never be zero,
so that the “limit point” behavior of (Sλ), i.e., dim N(λ) = 3n, should not be determined
by the equality R+(λ) = 0, as one would expect from the separated endpoints case in
Theorem 2.4.3. However, we can read Theorems 2.4.3, 2.4.8 and Corollary 2.4.10 also in
a different way. Namely, the number of linearly independent solutions of system (Sλ),
which are not square summable, is equal to dim Ker R+(λ). For the augmented system (Sλ)
we now have exactly the same statement, i.e., the number of linearly independent solutions
of system (Sλ), which are not square summable, is equal to dim Ker R+(λ).
Remark 3.3.8. The augmentation of system (Sλ) into the double dimension is a known
technique for studying the problems with jointly varying endpoints, see e.g. [17, 87, 88,
92,112,153]. The transformation introduced in this chapter has the advantage that it uses
the solutions z or Z of system (Sλ) rather than their components x, u or X, U as in the
above references. This yields a direct connection between the original system (Sλ) and
the augmented system (Sλ). For example, the boundary conditions in [153, Section 6] are
of the form
P1
(
−x0
xN+1
)
+ P2
(
u0
uN+1
)
= 0 (3.47)
with certain 2n × 2n matrices P1 and P2. One can see that the approach via (3.1) is
much easier and more transparent. The relationship between the transformation in the
above mentioned references and the transformation, which is utilized in this section, is
determined by the multiplication of the data (from one side or from both sides) by the
following 4n × 4n matrix
T :=


−I 0 0 0
0 0 I 0
0 I 0 0
0 0 0 I


= T−1
.
In particular, the equality
T


−x0
xN+1
u0
uN+1


=


x0
u0
xN+1
uN+1


=
(
z0
zN+1
)
gives a direct connection between the boundary conditions in (3.47) and (3.1).
3.4 Bibliographical notes
The results of this chapter were established in [A13] and their generalization to symplectic
systems on time scales was given in [A16, Section 8]. In addition, Corollary 3.2.3 is
published for the first time in the present setting and it is derived as a special case
of [A16, Corollary 8.5].
– 54 –
Chapter 4
Invariance of limit circle case for
two discrete systems
The question of the ultimate foundations and the ultimate
meaning of mathematics remains open; we do not know in
what direction it will find its final solution or even whether
a final objective answer can be expected at all. “Mathematizing”
may well be a creative activity of man, like language
or music, of primary originality, whose historical decisions
defy complete objective rationalization.
Hermann Weyl, see [106, pg. 319]
In this chapter we derive an invariance of the situation, when all solutions are square
summable, i.e., of the limit circle case for system (Sλ) as it was already stated in Theorem
2.4.17, see Remark 4.2.4. However, instead of system (Sλ) we consider two discrete
systems of the first order in the form
ˆzk+1(λ) =
(
Sk + λVk
)
ˆzk(λ), (Sλ)
˜zk+1(λ) =
(
Sk + λVk
)
˜zk(λ), (Sλ)
where k ∈ [0, ∞)Z, λ ∈ C, and S, V, S, V ∈ C([0, ∞)Z)2n×2n. Moreover, the coefficients of
systems (Sλ) and (Sλ) satisfy for every k ∈ [0, ∞)Z the relations
S
∗
k JSk = J, V
∗
k JSk + S
∗
k JVk = 0, V
∗
k JVk = 0, (4.1)
k := JVk JS
∗
k J ≥ 0, (4.2)
i.e., the matrix k is Hermitian and positive semidefinite for all k ∈ [0, ∞)Z, compare
with Remark 4.1.4(i). Despite the analogy between the conditions in (4.1), (4.2) and the
assumptions for system (Sλ) displayed in (2.1), we emphasize that systems (Sλ) and (Sλ)
are generally non-symplectic in the sense of the terminology introduced in Chapter 2, see
also Remark 4.1.4(ii).
Our investigation is motivated by Walker’s results in [168], where an analogous problem
was studied for a pair of the non-Hermitian linear Hamiltonian differential systems
ˆz′
(t, λ) = [H(t) + λW(t)] ˆz(t, λ), (H
R
λ )
˜z′
(t, λ) = [JH∗
(t)J + λW(t)] ˜z(t, λ) (H
R
λ )
with t ∈ [a, ∞) and H(t), W(t) being locally integrable 2n × 2n complex matrix-valued
functions such that the matrix W(t) is Hamiltonian and −JW(t) ≥ 0 on [a, ∞). More
– 55 –
Chapter 4. Invariance of limit circle case for two discrete systems
specifically, let us denote by L2
A
the space of all functions z : [a, ∞) → C2n, which are square
integrable with respect to the weight A(t) := −JW(t), i.e., −
∫ ∞
a
z∗(t)JW(t)z(t)dt < ∞. Then
it was proven in [168, Theorem 2] that if
∫ ∞
a
| tr W(t)|dt < ∞ (4.3)
and all solutions of systems (H
R
λ0
) and (H
R
λ0
) belong to L2
A
for some λ0 ∈ C, then this
property holds for all solutions of systems (H
R
λ ) and (H
R
λ ) with an arbitrary λ ∈ C.
This statement extends the invariance of the limit circle case for one system (H
R
λ ) with the
coefficient matrix H(t) being Hamiltonian on [a, ∞), i.e., for the situation (H
R
λ )=(H
R
λ )=(2.5),
established by Atkinson in [9, Theorem 9.11.2]. Although the main result of this chapter
(Theorem 4.2.2) yields a discrete counterpart of [168, Theorem 2], we point out that,
surprisingly, it does not require any analogue of condition (4.3).
4.1 Preliminaries
In this section we collect some auxiliary results about the coefficients of systems (Sλ)
and (Sλ). Similarly as in Chapter 2, let us define
5k(λ) := Sk + λVk, 5k(λ) := Sk + λVk. (4.4)
Then the identities in (4.1) imply for all k ∈ [0, ∞)Z and λ ∈ C that
5∗
k(λ)J5k(¯λ) = J. (4.5)
Thus from the invertibility of J it follows immediately that the matrices 5k(λ) and 5k(λ)
are invertible for any k ∈ [0, ∞)Z and λ ∈ C with
5−1
k (λ) = −J5∗
k(¯λ)J, 5−1
k (λ) = −J5∗
k(¯λ)J. (4.6)
Hence any initial value problem associated with system (Sλ) or (Sλ) possesses a unique
solution on [0, ∞)Z. Moreover, the fundamental matrices of systems (Sλ) and (Sλ) are
invertible on the whole discrete interval [0, ∞)Z.
In the following lemma we give several conditions which are equivalent to (4.1).
Namely, we show that any variation of the superscripts star, hat, and tilde is possible.
Lemma 4.1.1. Let n ∈ N be given. For any k ∈ [0, ∞)Z the following conditions are equivalent.
(i) The matrices S(t), S(t), V(t), V(t) satisfy (4.1).
(ii) The matrices 5k(λ) and 5k(λ) satisfy (4.5) for all λ ∈ C.
(iii) The matrices S(t), S(t), V(t), V(t) satisfy
Sk JS
∗
k = J, Vk JS
∗
k + Sk JV
∗
k = 0, Vk JV
∗
k = 0. (4.7)
(iv) The matrices 5k(λ) and 5k(λ) satisfy for all λ ∈ C that
5k(λ)J5∗
k(¯λ) = J. (4.8)
Proof. The equivalence of (i) and (ii), and of (iii) and (iv), follows by direct calculations
with the notation introduced in (4.4). The equivalence of (ii) and (iv) is a consequence of
the relations in (4.6) and of the fact 5k(λ)5−1
k (λ) = I. ■
– 56 –
4.1. Preliminaries
Further conditions which are equivalent to (i)–(iv) in Lemma 4.1.1 can be obtained by
the conjugate transpose of the identities in (4.1), (4.5), (4.7), and (4.8).
Now we focus on the coefficients of systems (Sλ) and (Sλ) with different values of the
spectral parameter. Together with k defined in (4.2) we consider also the matrix
k := JVk JS
∗
k J. (4.9)
Lemma 4.1.2. Let n ∈ N be fixed and k ∈ [0, ∞)Z be such that the conditions in (4.1) hold. Then
the matrices k, k defined in (4.2) and (4.9), respectively, satisfy
k = ∗
k, k J k = 0, k J k = 0, (4.10)
and for all λ, ν ∈ C we have
J5k(λ)J5∗
k(ν)J = (λ − ¯ν) k − J, (4.11)
J5k(λ)J5∗
(ν)J = (λ − ¯ν) k − J. (4.12)
Proof. The above identities follow by direct calculations. ■
Identities (4.11) and (4.12) play a crucial role in the proof of the following generalization
of the Lagrange identity for two systems, compare with Theorem 2.1.7.
Theorem 4.1.3 (Generalized Lagrange identity). Let n, m ∈ N and λ, ν ∈ C be fixed and
assume that the conditions in (4.1) hold for all k ∈ [0, ∞)Z. If Z(λ), Z(ν) ∈ C([0, ∞)Z)2n×m solve
systems (Sλ) and (Sν) on [0, ∞)Z, respectively, then for any k ∈ [0, ∞)Z we have
[
Z
∗
k(λ)JZk(ν)
]
= (¯λ − ν)Z
∗
k+1(λ) k Zk+1(ν), (4.13)
[
Z
∗
k(λ)JZk(ν)
]
= (¯λ − ν)Z
∗
k+1(λ) k Zk+1(ν). (4.14)
Proof. Identity (4.13) follows from the first equality in (4.6) and from (4.11), because
[
Z
∗
k(λ)JZk(ν)
]
= Z
∗
k+1(λ)
[
J − 5∗−1
k (λ)J5−1
k (ν)
]
Zk+1(ν)
(4.6)
= Z
∗
k+1(λ)
[
J + J5k(¯λ)J5∗
k(¯ν)J
]
Zk+1(ν)
(4.11)
= (¯λ − ν)Z
∗
k+1(λ) k Zk+1(ν).
Similarly we get identity (4.14) from the second equality in (4.6) and from (4.12). ■
Remark 4.1.4.
(i) The results in Theorem 4.1.3 imply that in order to have a single weight matrix
for the semi-inner product and the semi-norm in the associated space of square
summable solutions, we must necessarily assume that k = k. This means, in
view of Lemma 4.1.2, that we need to have k Hermitian. This is, in fact, the
original motivation for our assumption (4.2).
(ii) In the continuous case it is obvious that systems (H
R
λ ) and (H
R
λ ) coincide if and only
if H(t) is Hamiltonian on [a, ∞). Now we give the answer to the same question for
systems (Sλ) and (Sλ). From the first equality in (4.1) (or from (4.6) with λ = 0) and
from the first and the second conditions in (4.1) we obtain, respectively,
Sk = −JS
∗−1
k J and Vk = JS
∗−1
k V
∗
k S
∗−1
k J. (4.15)
– 57 –
Chapter 4. Invariance of limit circle case for two discrete systems
Therefore, S(t) = S(t) in (4.15) if and only if
S
∗
k JSk = J,
i.e., Sk is symplectic for all k ∈ [0, ∞)Z. In this case, Vk = Vk in (4.15) if and only if the
matrix Vk JS
∗
k = Vk JS
∗
k is Hermitian for all k ∈ [0, ∞)Z, i.e., the matrix k is Hermitian
on [0, ∞)Z. In other words, systems (Sλ) and (Sλ) satisfying (4.1) and (4.2) coincide
if and only if they represent the discrete symplectic system studied in Chapter 2,
i.e., system (Sλ) with the coefficient matrices satisfying (2.1).
Finally, we calculate the determinant of the matrices 5k(λ), 5k(λ) and of their product.
Let k ∈ [0, ∞)Z be such that the conditions in (4.1) hold. From the first equality in (4.1) we
obtain for any λ ∈ C that
5k(λ) = (I + λJ k)Sk and 5k(λ) = (I + λJ k)Sk.
Moreover, by the second and third identities in (4.10) the matrices λJ k and λJ k are
nilpotent of degree two, which with the aid of Proposition 1.1.3 yields
det(I + λJ k) = 1 = det(I + λJ k).
Therefore det 5k(λ) = det Sk and det 5k(λ) = det Sk, i.e., the determinants do not depend
on λ. Consequently from the first condition in (4.1) we get
det 5∗
k(λ)J5k(λ) = det S
∗
k JSk
(4.1)
= det J = 1, i.e., det 5∗
k(λ) × det 5k(λ) = 1. (4.16)
In addition, from the latter equality and Remark 4.1.4(ii) one concludes that when systems
(Sλ) and (Sλ) coincide, then the absolute value of det 5k(λ) is equal to one, as we
claim in Lemma 2.1.3.
4.2 Main result
In this section we establish the main result of this chapter (Theorem 4.2.2). We also provide
sufficient conditions for the invariance, present an illustrative example, and discuss some
special cases. For convenience, we summarize the used notation.
Notation 4.2.1. The number n ∈ N is fixed and S, S, V, V, ∈ C([0, ∞)Z)2n×2n are such that
the conditions in (4.1) and (4.2) are satisfied for all k ∈ [0, ∞)Z.
It follows from Remark 4.1.4(i) that with Notation 4.2.1 we have just one weight matrix
k = k, which is Hermitian and positive semidefinite on [0, ∞)Z. Therefore, we denote
by ℓ2 the space of all sequences defined on [0, ∞)Z, which are square summable with
respect to the weight matrix k, i.e.,
ℓ2
[0, ∞)Z = ℓ2
:=
{
z ∈ C([0, ∞)Z)2n
∞∑
k=0
z∗
k k zk < ∞
}
.
Let us note that the statement in Theorem 4.2.2 below remains the same for other types
of unbounded discrete intervals, such as for (−∞, b]Z or (−∞, ∞)Z, if the corresponding
space ℓ2 is defined over that interval.
Theorem 4.2.2. Let us assume that there exists λ0 ∈ C such that all solutions of systems (Sλ0
)
and (Sλ0
) belong to ℓ2 . Then all solutions of systems (Sλ) and (Sλ) belong to ℓ2 for any λ ∈ C.
– 58 –
4.2. Main result
Proof. Let λ0 ∈ C be as stated in the theorem and λ ∈ CK{λ0} be fixed. For ν ∈ {λ, λ0} we
denote by k(ν) and k(ν) the fundamental matrices of systems (Sν) and (Sν), respectively,
such that 0(ν) = I = 0(ν). In the first part of the proof we show that all solutions of
system (Sλ) belong to ℓ2 . Since the matrices k(λ) and k(λ0) are obviously invertible on
[0, ∞)Z, it follows that for every k ∈ [0, ∞)Z we have
k(λ) = k(λ0) k (4.17)
for some invertible matrix k ∈ C2n×2n, i.e., k = −1
k (λ0) k(λ). Hence by straightforward
calculations with using (4.1), (4.4), (4.6), and (4.17) we get
k
(4.17)
= −1
k+1(λ0)
[
5k(λ) − 5k(λ0)
]
k(λ)
(4.4)
= (λ − λ0) −1
k+1(λ0)Vk k(λ0) k
= (λ − λ0) −1
k+1(λ0)Vk 5−1
k (λ0) k+1(λ0) k
(4.6)
= −(λ − λ0) −1
k+1(λ0)Vk J5∗
k(¯λ0)J k+1(λ0) k
(4.1)
= −(λ − λ0) −1
k+1(λ0)Vk JS
∗
k J k+1(λ0) k
= (λ − λ0) −1
k+1(λ0)J k k+1(λ0) k. (4.18)
It means that k satisfies the recurrence relation
k+1 =
[
I + (λ − λ0)ϒk
]
k, k ∈ [0, ∞)Z, (4.19)
where ϒ ∈ C([0, ∞)Z)2n×2n is given by the formula
ϒk := −1
k+1(λ0)J k k+1(λ0) = −
[ ∗
k+1(λ0)J k+1(λ0)
]−1 ∗
k+1(λ0) k k+1(λ0). (4.20)
Identity (4.17) implies that for the required conclusion it suffices to prove the boundedness
of || k ||σ on [0, ∞)Z. However equality (4.19) and the submultiplicative property of the
norm ||·||σ yield
k+1 σ
=
[
I + (λ − λ0)ϒk
]
k
σ
≤
(
||I||σ + |λ − λ0 | × ||ϒk ||σ
)
× k σ
=
(
1 + |λ − λ0 | × ||ϒk ||σ
)
×
[
I + (λ − λ0)ϒk−1
]
k−1
σ
≤
(
1 + |λ − λ0 | × ||ϒk ||σ
)
×
(
1 + |λ − λ0 | × ||ϒk−1 ||σ
)
× k−1 σ
≤ · · · ≤
(
1 + |λ − λ0 | × ||ϒk ||σ
)
× · · · ×
(
1 + |λ − λ0 | × ||ϒ0 ||σ
)
× 0 σ
≤ e|λ−λ0 |ωk with ωk :=
k∑
j=0
||ϒj ||σ, (4.21)
where in the last step we used the inequality 1 + x ≤ ex and the fact 0 = I; cf. Proposition
1.1.4. Therefore we need to show that limk→∞ ωk < ∞.
Since k ≥ 0 on [0, ∞)Z, we obtain from the Cauchy–Schwarz inequality and the
arithmetic-geometric mean inequality for any ξ, ζ ∈ C2n that
ξ∗
kζ ≤
(
ξ∗
kξ
)1/2 (
ζ∗
kζ
)1/2
≤ 1
2
(
ξ∗
kξ + ζ∗
kζ
)
,
which for any ˜z, ˆz ∈ ℓ2 implies
∞∑
k=0
˜z∗
k k ˆzk ≤
∞∑
k=0
˜z∗
k k ˆzk ≤ 1
2
∞∑
k=0
(
˜z∗
k k ˜zk + ˆz∗
k k ˆzk
)
< ∞.
– 59 –
Chapter 4. Invariance of limit circle case for two discrete systems
Hence inequality (1.7) and the assumption that all solutions of systems (Sλ0
) and (Sλ0
)
belong to ℓ2 , yield
∞∑
k=0
∗
k+1(λ0) k k+1(λ0) σ
(1.7)
≤
∞∑
k=0
∗
k+1(λ0) k k+1(λ0) 1
≤ ε < ∞ (4.22)
for some ε > 0. Now we put
Qk := ∗
k+1(λ0)J k+1(λ0)
and show that the value of Q−1
k σ
is bounded on [0, ∞)Z. By the generalized Lagrange
identity in Theorem 4.1.3 we get
Qk = J − 2i im(λ0)
∞∑
k=0
∗
j+1(λ0) j j+1(λ0),
and so inequality (4.22) implies that the limit limk→∞ Qk exists and is bounded in the
spectral norm, i.e., ||Qk ||σ is bounded on [0, ∞)Z. Therefore also the adjugate matrix Q
adj
k
is bounded on [0, ∞)Z in the spectral norm. Moreover, it holds
det Qk = det ∗
k+1(λ0) × det k+1(λ0) = det ∗
k(λ0) × det k(λ0) × det 5∗
k(λ0) × det 5k(λ0)
(4.16)
= det ∗
k−1(λ0) × det k−1(λ0) × det 5∗
k−1(λ0) × det 5k−1(λ0)
= · · · = det ∗
0(λ0) × det 0(λ0) = 1,
which yields Q−1
k
= Q
adj
k
and consequently the matrices Q−1
k
are bounded in the spectral
norm, i.e., Q−1
k σ
≤ κ for all k ∈ [0, ∞)Z and some κ > 0. By combining the submultiplicative
property of the spectral norm, (4.20), and (4.22) we get
∞∑
k=0
||ϒk ||σ ≤
∞∑
k=0
Q−1
k σ
× ∗
k+1(λ0) k k+1(λ0) σ
≤ κε < ∞,
i.e., limk→∞ ωk < ∞ by (4.21). Thus k σ
≤ τ on [0, ∞)Z for some τ > 0. In turn,
the definition of k, the second inequality in (1.7), and the submultiplicativity and selfadjointness
of the spectral norm imply
(2n)−3/2
∞∑
k=0
∗
k+1(λ) k k+1(λ) 1
(1.7)
≤
∞∑
k=0
∗
k+1(λ) k k+1(λ) σ
(4.17)
=
∞∑
k=0
∗
k+1
∗
k+1(λ0) k k+1(λ0) k+1 σ
≤
∞∑
k=0
k+1
2
σ
× ∗
k+1(λ0) k k+1(λ0) σ
≤ τ2
∞∑
k=0
∗
k+1(λ0) k k+1(λ0) σ
< ∞,
because all columns of the fundamental matrix (λ0) belong to ℓ2 . This shows that all
columns of (λ) belong to ℓ2 and consequently any solution of system (Sλ) is square
summable with respect to k. For the proof of the fact that all solutions of system (Sλ)
– 60 –
4.2. Main result
are also in ℓ2 we only switch the roles of systems (Sλ) and (Sλ). Namely, we define
k := −1
k (λ0) k(λ) and similarly as in (4.18) we derive
k = (λ − λ0) −1
k+1(λ0)J k k+1(λ0) k.
But since we have k = k for all k ∈ [0, ∞)Z, the rest of the proof is the same as in the
previous part. ■
In the following result we give sufficient conditions in terms of the coefficient matrices,
which guarantee that all solutions of systems (Sλ) and (Sλ) belong to ℓ2 for any λ ∈ C.
Let us note that the matrix norm ||·||1 used in (4.23) below can be replaced by any other
matrix norm because of their equivalence.
Corollary 4.2.3. Let us assume that
∞∑
k=0
Sk − I 1
< ∞,
∞∑
k=0
Sk − I 1
< ∞, and
∞∑
k=0
k 1
< ∞. (4.23)
Then all solutions of systems (Sλ) and (Sλ) belong to ℓ2 for any λ ∈ C.
Proof. It suffices to show that the assumptions of Theorem 4.2.2 are satisfied for λ0 = 0.
The equality 5k(0) = Sk for all k ∈ [0, ∞)Z and the first condition in (4.23) imply by Proposition
1.1.4 that for a fundamental matrix (0) ∈ C([0, ∞)Z)2n×2n of system (S0) there exists
κ > 0 such that k(0) 1
≤ κ < ∞ for all k ∈ [0, ∞)Z. Hence by the submultiplicativity and
self-adjointness of the matrix norm ||·||1 and the third condition in (4.23) we have
∞∑
k=0
∗
k+1(0) k k+1(0) 1
≤ κ2
∞∑
k=0
k 1
< ∞,
i.e., all solutions of system (S0) belong to ℓ2 . In a similar way we prove that any solution of
system (S0) also belongs to ℓ2 . Thus Theorem 4.2.2 implies that all solutions of systems (Sλ)
and (Sλ) are in ℓ2 for any λ ∈ C. ■
Remark 4.2.4. In accordance with Remark 4.1.4(ii), if both systems (Sλ) and (Sλ) coincide,
Theorem 4.2.2 reduces to Theorem 2.4.17 and Corollary 4.2.3 to Corollary 2.4.22. Therefore,
with a slight abuse in the terminology, Theorem 4.2.2 can be interpreted as the invariance
of the limit circle case for systems (Sλ) and (Sλ), i.e., the situation when all solutions of
systems (Sλ) and (Sλ) belong to ℓ2 for any λ ∈ C.
Now we provide an illustrative example of the established invariance, i.e., the application
of Theorem 4.2.2. This example also shows that all solutions of systems (Sλ)
and (Sλ) may be in ℓ2 for any λ ∈ C even when conditions (4.23) in Corollary 4.2.3 are not
satisfied.
Example 4.2.5. Let n = 1 and fix ε ∈ R, ε ≥ 1. Let {vk}∞
k=0
be a real sequence such that
vk ≥ 0 for all k ∈ [0, ∞)Z and
∑∞
k=0 ε2k vk < ∞. We note that then the series
∑∞
k=0 vk < ∞
and
∑∞
k=0 vk/ε2k < ∞ are convergent as well, because 0 ≤ vk/ε2k ≤ vk ≤ ε2k vk. Consider
systems (Sλ) and (Sλ) with
Sk := (1/ε)I, Vk :=
(
0 vk
0 0
)
, Sk := εI, Vk :=
(
0 ε2 vk
0 0
)
, k :=
(
0 0
0 εvk
)
(4.24)
– 61 –
Chapter 4. Invariance of limit circle case for two discrete systems
for all k ∈ [0, ∞)Z. Then all the conditions (4.1) and (4.2) are satisfied. The fundamental
matrices k(0) =
(
ˆz[1]
k
(0), ˆz[2]
k
(0)
)
and k(0) =
(
˜z[1]
k
(0), ˜z[2]
k
(0)
)
of systems (S0) and (S0)
with (4.24) satisfying 0(0) = I = 0(0) are equal to k(0) = (1/εk)I and k(0) = εk I, so
that ˆz[1]
k
(0) = (1/εk, 0)⊤, ˆz[2]
k
(0) = (0, 1/εk)⊤, ˜z[1]
k
(0) = (εk, 0)⊤, and ˜z[2]
k
(0) = (0, εk)⊤ for all
k ∈ [0, ∞)Z. Then with the notation ||z||2
:=
∑∞
k=0 z∗
k k zk we have
ˆz[1]
(0)
2
= 0, ˆz[2]
(0)
2
= ε
∞∑
k=0
vk/ε2k
< ∞,
˜z[1]
(0)
2
= 0, ˜z[2]
(0)
2
= ε
∞∑
k=0
ε2k
vk < ∞,
i.e., the solutions ˆz[1]
(0), ˆz[2]
(0), ˜z[1]
(0), ˜z[2]
(0) belong to ℓ2 . Thus the assumptions of Theorem
4.2.2 are satisfied for λ0 = 0, which implies that all solutions of systems (Sλ) and (Sλ)
with the coefficients specified in (4.24) belong to ℓ2 for any λ ∈ C. Indeed, for any λ ∈ C
the fundamental matrices k(λ) and k(λ) of systems (Sλ) and (Sλ) with (4.24) satisfying
0(λ) = I = 0(λ) are given by
k(λ) =


1/εk (λ/εk−1)
∑k−1
j=0 vj
0 1/εk

 , k(λ) =


εk λεk+1
∑k−1
j=0 vj
0 εk

 ,
from which we obtain again
ˆz[1]
(λ)
2
= 0 = ˜z[1]
(λ)
2
and ˆz[2]
(λ)
2
< ∞, ˜z[2]
(λ)
2
< ∞.
One also easily observe that the first two conditions in (4.23) are not satisfied (since
the corresponding series are divergent), but still all solutions of systems (Sλ) and (Sλ)
with (4.24) do belong to ℓ2 . In addition, we note that for ε = 1 both systems coincide and
reduce to the system investigated in Example 2.5.3, see (2.78). ▲
Finally, let us consider systems (Sλ) and (Sλ), which correspond to the following
n-vector-valued difference equations of order 2m and of the Sturm–Liouville type
m∑
s=0
(−1)s s
[
P
[s]
k
s
ˆyk+1−s(λ)
]
= λWk ˆyk+1(λ), (Eλ)
m∑
s=0
(−1)s s
[
P
[s]
k
s
˜yk+1−s(λ)
]
= λWk ˜yk+1(λ), (Eλ)
on [0, ∞)Z, where P
[0]
, . . . , P
[m]
, P
[0]
, . . . , P
[m]
, W, W ∈ C([0, ∞)Z)n×n with det P
[m]
k 0 and
det P
[m]
k 0 for all k ∈ [0, ∞)Z. In particular, if m = n = 1 we obtain the pair of scalar
difference equations
−
(
p[1]
k
ˆyk(λ)
)
+ p[0]
k
ˆyk+1(λ) = λwk ˆyk+1(λ),
−
(
p[1]
k
˜yk(λ)
)
+ p[0]
k
˜yk+1(λ) = λwk ˜yk+1(λ),
where p[0]
, p[1]
, p[0]
, p[1]
, w, w ∈ C([0, ∞)Z) with p[1]
0 and p[1]
0 for all k ∈ [0, ∞)Z. Then
equations (Eλ) and (Eλ) can be written, respectively, as systems (Sλ) and (Sλ) with the
coefficients given similarly as in (1.20)–(1.22) and (2.11), which yield
k = diag
{
Wk, 0, . . . , 0
}
, k = diag
{
Wk, 0, . . . , 0
}
. (4.25)
– 62 –
4.3. Bibliographical notes
The first and second conditions in (4.1) imply P
[j]
k
= P
[j]∗
k
and Wk = W
∗
k for all j = 0, . . . , m
and k ∈ [0, ∞)Z, while the third condition is satisfied trivially. Moreover, assumption (4.2)
forces that Wk = Wk ≥ 0 on [0, ∞)Z, i.e., the weight matrices Wk and Wk are Hermitian
matrices and coincide on the interval [0, ∞)Z. With the vectors ˆzk(λ), ˜zk(λ) defined similarly
as in (1.20) and the matrices k, k from (4.25) we have
ˆz∗
k+1(λ) k ˆzk+1(λ) = ˆy∗
k+1(λ)Wk ˆyk+1(λ) and ˜z∗
k+1(λ) k ˜zk+1(λ) = ˜y∗
k+1(λ)Wk ˜yk+1(λ).
This shows that the associated space of square summable sequences has the form
ℓ2
W
:=
{
y ∈ C([0, ∞)Z)n
∞∑
k=0
y∗
k+1(λ)Wk yk+1(λ) < ∞
}
.
Then from Theorem 4.2.2 we get the following result, which in the scalar case provides
a discrete analogue of [168, Theorem 1].
Corollary 4.2.6. Let the numbers m, n ∈ N be given and P
[0]
, . . . , P
[m]
, W ∈ C([0, ∞)Z)n×n be
such that Wk = W
∗
k ≥ 0 on [0, ∞)Z. Consider equations (Eλ) and (Eλ) with P
[j]
k
:= P
[j]∗
k
and
Wk := Wk for all j ∈ {0, . . . , m} and k ∈ [0, ∞)Z. If there exists λ0 ∈ C such that all solutions of
equations (Eλ0
) and (Eλ0
) belong to ℓ2
W
, then all solutions of equations (Eλ) and (Eλ) belong to the
space ℓ2
W
for an arbitrary λ ∈ C.
Remark 4.2.7. If, in addition, the coefficient matrices P
[0]
, . . . , P
[m]
are Hermitian, then
from Corollary 4.2.6 one easily concludes the invariance of the limit circle case for any
even order vector-valued Sturm–Liouville difference equation discussed in Remark 2.4.18.
Similarly we can derive also the invariance for pairs of difference/discrete equations of
the type as in (2.9) or (2.10).
4.3 Bibliographical notes
The results of this chapter are a special case of the invariance of the limit circle case for
two differential systems on time scales established in [A19] but without the shift in the
definition of the space ℓ2 . The present reformulation and the proof are published for
the first time in the setting of systems (Sλ) and (Sλ). The statement of Corollary 4.2.6 is
new in the case n > 1 or m > 1. Moreover, the presence of the shift in the definition of
ℓ2 produces less restrictive assumptions on the coefficients of equations (Eλ) and (Eλ),
compare Corollary 4.2.6 with m = n = 1 and [A19, Corollary 4.4] with T = Z. Finally,
Example 4.2.5 corresponds to [A19, Example 4.6].
– 63 –
Chapter 4. Invariance of limit circle case for two discrete systems
– 64 –
Chapter 5
Polynomial and analytic
dependence on spectral parameter
It can be of no practical use to know that π is irrational, but
if we can know, it surely would be intolerable not to know.
Edward Charles Titchmarsh, see [136, pg. 113]
In this chapter we extend some of the previous results to systems with polynomial
or analytic dependence on the spectral parameter. More specifically, we consider the
discrete symplectic system
zk+1(λ) = Sk(λ)zk(λ), (Sλ)
whose coefficient matrix S(λ) ∈ C([0, ∞)Z)2n×2n is analytic (or in a special case only polynomial)
in the spectral parameter λ ∈ C in a neighborhood of 0, i.e.,
Sk(λ) =
∞∑
j=0
λj
S
[j]
k
, (5.1)
and it satisfies the symplectic-type identity
S∗
k(¯λ)JSk(λ) = J. (5.2)
System (Sλ) includes several significant special cases known in the literature. Obviously,
system (Sλ) from Chapter 2 is a special case of (Sλ), see (2.2). Furthermore, we will
see that the linear Hamiltonian difference system in (2.6) leads to system (Sλ) with polynomial
dependence on λ. Therefore, in order to unify the known theory of systems (2.6)
and (Sλ), it is necessary to study the systems with polynomial dependence on λ. In fact,
our interest in system (Sλ) is motivated by the latter observation and also by [49], where
system (Sλ) with S[0]
k
≡ I was investigated.
We discuss an eigenvalue problem associated with system (Sλ) and develop the theory
of Weyl disks and square summable solutions (including the limit point and limit circle
cases) for system (Sλ). However, we point out that in this treatment we encounter several
“problems”, which did not appear in Chapters 2–4, i.e., when the dependence on λ was
only linear. For example, the weight matrix is no longer constant in λ, which implies
that the validity of the crucial weak Atkinson condition may now depend on λ, see
– 65 –
Chapter 5. Polynomial and analytic dependence on spectral parameter
Hypotheses 2.3.7 and 5.3.2. Also the maximal number of linearly independent square
summable solutions (i.e., the limit circle case) is not any more invariant with respect
to λ ∈ C, see Theorem 2.4.17 and Example 5.3.9. On the other hand, we prove in
Theorems 5.4.1 and 5.4.5 that for system (Sλ) with a special quadratic dependence on λ
the invariance of the limit circle case holds true as in Chapter 4.
This chapter is organized as follows. In the next section we derive several preliminary
results on system (Sλ) and its coefficient matrix S(λ). We also prove a general form of the
Lagrange identity for system (Sλ) including the explicit calculation of the corresponding
weight matrix in terms of the coefficients of (Sλ), see Theorem 5.1.6. As a consequence
we obtain the J-monotonicity of a fundamental matrix of system (Sλ), which is used
in [49] for proving the Krein traffic rules for the eigenvalues of the fundamental matrix.
In Section 5.2 we discuss in more details some special cases of system (Sλ), which are
known in the literature. In Section 5.3 we show that under appropriate Atkinson-type
conditions involving the weight matrix, the theory of eigenvalues, Weyl disks, and square
summable solutions developed in Chapter 2 remains valid without any change also for
system (Sλ). Finally, in Section 5.4 we establish the invariance of the limit circle case
for system (Sλ) with a special quadratic dependence on the spectral parameter, which
includes also system (2.6) with Ek ≡ 0.
5.1 Preliminaries and Lagrange identity
Throughout this chapter we assume that Sk(λ) has a positive radius of convergence as
a power series with respect to λ uniformly in k ∈ [0, ∞)Z. It means that there exists ε > 0
such that Sk(λ) is absolute convergent for all λ ∈ C satisfying |λ| < ε and all k ∈ [0, ∞)Z. We
denote this region of convergence as CS, i.e., we have CS := {λ ∈ C | |λ| < ε}. Moreover,
we say that Sk(λ) is a polynomial matrix (of degree p) with respect to λ, if there exists p ∈ N0
such that Sk(λ) =
∑p
j=0
λjS
[j]
k
with S
[p]
k
0. Obviously, in the latter case we can take ε = ∞,
i.e., CS = C. If S
[j]
k
0 for infinitely many j ∈ N0, we say that the matrix Sk(λ) is analytic
with respect to λ.
Using the absolute convergence of the matrices Sk(λ) for all λ ∈ CS, identity (5.2) can
be equivalently written as
S[0]∗
k
JS[0]
k
= J and
m∑
j=0
S
[j]∗
k
JS
[m−j]
k
= 0 for all m ∈ N. (5.3)
If Sk(λ) is a polynomial matrix of degree p in λ, then the sum in (5.3) is nontrivial only for
m = 0, . . . , 2p. Moreover, identity (5.2) also implies that Sk(λ) is invertible with
S−1
k (λ) = −JS∗
k(¯λ)J = −
∞∑
j=0
λj
JS
[j]∗
k
J. (5.4)
The following lemma provides several equivalent formulations of assumption (5.2),
compare with Lemma 2.1.1. The proof follows (again) by direct calculations and from (5.4).
Lemma 5.1.1. Let n ∈ N be given. For any k ∈ [0, ∞)Z the following conditions are equivalent.
(i) The matrix Sk(λ) in (5.1) satisfies identity (5.2) for all λ ∈ CS.
(ii) The matrices S[0]
k
, S[1]
k
, . . . satisfy the equalities in (5.3).
– 66 –
5.1. Preliminaries and Lagrange identity
(iii) The matrix Sk(λ) satisfies
Sk(λ)JS∗
k(¯λ) = J for all λ ∈ CS. (5.5)
(iv) The matrices S[0]
k
, S[1]
k
, . . . satisfy
S[0]
k
JS[0]∗
k
= J and
m∑
j=0
S
[j]
k
JS
[m−j]∗
k
= 0 for all m ∈ N. (5.6)
Given Lemma 5.1.1 we can summarize the basic notation used in this chapter.
Notation 5.1.2. The number n ∈ N is fixed and S[0]
, S[1]
, · · · ∈ C([0, ∞)Z)2n×2n are such that
(i) the equalities in (5.3) are satisfied for all k ∈ [0, ∞)Z and
(ii) the radius of convergence of Sk(λ) in λ is equal to ε > 0 uniformly in k ∈ [0, ∞)Z.
The invertibility of Sk(λ) guarantees that system (Sλ) is uniquely solvable on [0, ∞)Z
for any initial value at any k0 ∈ [0, ∞)Z. Moreover, when Sk(λ) is polynomial of degree p
in λ, we obtain an additional information about its determinant, which generalizes the
result in Theorem 2.1.3. However, when Sk(λ) is analytic and not polynomial in λ, then
the following statement may be violated as we will demonstrate in Example 5.2.4.
Theorem 5.1.3. Let k ∈ [0, ∞)Z and λ ∈ CS be such that the matrix Sk(λ) is polynomial in λ.
Then | det Sk(λ)| = det S[0]
k
= 1.
Proof. If Sk(λ) is polynomial in λ, then identity (5.4) implies that Sk(λ) is even an unimodular
polynomial matrix. Thus, its determinant is constant in λ and we have
| det Sk(λ)| = | det Sk(0)| = det S[0]
k
= 1,
because S[0]
k
is a symplectic matrix by the first equality in (5.3). ■
Now we return to a general case of the analytic dependence on λ. The following
lemma provides an extension of identity (5.5) and it is a main tool for the proof of the
Lagrange identity given below.
Lemma 5.1.4. For all k ∈ [0, ∞)Z and any λ, ν ∈ CS we have
Sk(λ)JS∗
k(ν) = J + (λ − ¯ν)Λk(λ, ¯ν), (5.7)
where the matrix Λ(λ, ¯ν) ∈ C([0, ∞)Z)2n×2n is defined by
Λk(λ, ¯ν) :=
∞∑
m=0
m∑
j=0
λm−j
¯νj
j∑
ℓ=0
S[m−ℓ+1]
k
JS[ℓ]∗
k
. (5.8)
Moreover, for ν = λ the matrix Λk(λ, ¯λ) is Hermitian for all k ∈ [0, ∞)Z.
Remark 5.1.5. If Sk(λ) is a polynomial matrix of degree p in λ, then the infinite sum in (5.8)
is in fact a finite sum for m = 0, . . . , 2p − 1. Observe also that identity (5.8) reduces to (5.5)
when ν = ¯λ. Moreover, we point out that the Hermitian property of Λk(λ, ¯λ) was already
shown in [49, Proposition 1].
– 67 –
Chapter 5. Polynomial and analytic dependence on spectral parameter
Proof of Lemma 5.1.4. Let k ∈ [0, ∞)Z and λ, ν ∈ CS be fixed. The power series for Sk(λ) and
S∗
k
(ν) converge absolutely, so that the terms in the product Sk(λ)JS∗
k
(ν) can be re-arranged
to the separate powers of λm−j ¯νj, that is,
Sk(λ)JS∗
k(ν) =
∞∑
m=0
m∑
j=0
λm−j
¯νj
S
[m−j]
k
JS
[j]∗
k
.
By using identity (5.6) for each m ∈ N, we replace the term ¯νm S[0]
k
JS[m]∗
k
by
−¯νm
(
S[m]
k
JS[0]∗
k
+ S[m−1]
k
JS[1]∗
k
+ · · · + S[1]
k
JS[m−1]∗
k
)
.
Thus, with the aid of the first identity in (5.6), we get
Sk(λ)JS∗
k(ν) = J +
∞∑
m=1
m∑
j=1
(λj
− ¯νj
) ¯νm−j
S
[j]
k
JS
[m−j]∗
k
.
Upon factoring λ − ¯ν out of each term λj − ¯νj = (λ − ¯ν)
∑j
ℓ=1
λj−ℓ ¯νℓ−1 and collecting the
remaining products with the same powers of λ and ¯ν, we obtain
Sk(λ)JS∗
k(ν) = J + (λ − ¯ν)
∞∑
m=1
m∑
j=1
( j∑
ℓ=1
λj−ℓ
¯νℓ−1
)
¯νm−j
S
[j]
k
JS
[m−j]∗
k
= J + (λ − ¯ν)
∞∑
m=0
m∑
j=0
( j∑
ℓ=0
λj−ℓ
¯νm+ℓ−j
)
S
[j+1]
k
JS
[m−j]∗
k
= J + (λ − ¯ν)
∞∑
m=0
m∑
j=0
λm−j
¯νj
j∑
ℓ=0
S[m−ℓ+1]
k
JS[ℓ]∗
k
= J + (λ − ¯ν)Λk(λ, ¯ν),
where we used also the formula
∑m
j=0
∑j
ℓ=0
aj,ℓ =
∑m
ℓ=0
∑m
j=ℓ aj,ℓ. Finally, for ν := λ we get
from the fact J∗ = −J and identities (5.6) that the matrix Λk(λ, ¯λ) is Hermitian. ■
The following theorem represents the main result of this section. Its relationship to
known discrete Lagrange identities in the literature is discussed in Section 5.2.
Theorem 5.1.6 (Lagrange identity). Let the numbers m ∈ N and λ, ν ∈ CS be given. If the
sequences Z(λ), Z(ν) ∈ C([0, ∞)Z)2n×m solve systems (Sλ) and (Sν) on [0, ∞)Z, respectively, then
for any k ∈ [0, ∞)Z we have
[
Z∗
k(λ)JZk(ν)
]
= (¯λ − ν)Z∗
k+1(λ)JΛj(¯λ, ν)JZk+1(ν), (5.9)
Z∗
k+1(λ)JZk+1(ν) = Z∗
0(λ)JZ0(ν) + (¯λ − ν)
k∑
j=0
Z∗
j+1(λ)JΛj(¯λ, ν)JZj+1(ν). (5.10)
In particular, for ν = ¯λ and ν = λ we have on [0, ∞)Z, respectively,
Z∗
k(λ)JZk(¯λ) ≡ Z∗
0(λ)JZ0(¯λ), (5.11)
Z∗
k+1(λ)JZk+1(λ) = Z∗
0(λ)JZ0(λ) − 2i im(λ)
k∑
j=0
Z∗
j+1(λ)JΛj(¯λ, λ)JZj+1(λ). (5.12)
– 68 –
5.1. Preliminaries and Lagrange identity
Proof. Given that Zk(λ) and Zk(ν) satisfy systems (Sλ) and (Sν) for all k ∈ [0, ∞)Z, respectively,
we obtain from formula (5.4) and Lemma 5.1.4 that
[
Z∗
k(λ)JZk(ν)
]
= Z∗
k+1(λ)
[
J − S∗−1
k (λ)JS−1
k (ν)
]
Zk+1(ν)
(5.4)
= Z∗
k+1(λ)
[
J + JSk(¯λ)JS∗
k(¯ν)J
]
Zk+1(ν)
(5.7)
= (¯λ − ν)Z∗
k+1(λ)JΛk(¯λ, ν)JZk(ν).
Identities (5.9)–(5.12) are only direct consequences of (5.9). ■
Identity (5.12) indicates that the matrix JΛk(¯λ, λ)J will play an important role in the
study of square summable solutions of system (Sλ), especially in the definition of the
semi-inner product associated with system (Sλ), see Section 5.3. Hence we define the
Hermitian 2n × 2n matrix
Ψk(λ) := JΛk(¯λ, λ)J =
∞∑
m=0
m∑
j=0
¯λm−j
λj
j∑
ℓ=0
JS[m−ℓ+1]
k
JS[ℓ]∗
k
J. (5.13)
Remark 5.1.7. The expression of the matrix Ψk(λ) given in (5.13) can be significantly
simplified for real λ, i.e., for λ ∈ CS ∩ R. In particular, if we denote by ˙Sk(λ) := d
dλ Sk(λ)
the derivative of Sk(λ) with respect to λ, then
Ψk(λ) =
∞∑
m=0
m∑
j=0
λm
j∑
ℓ=0
JS[m−ℓ+1]
k
JS[ℓ]∗
k
J = J ˙Sk(λ)JS∗
k(λ)J = −JSk(λ)J ˙S∗
k(λ)J, (5.14)
where the last equality follows from the fact Ψ∗
k
(λ) = Ψk(λ). The matrix J ˙Sk(λ)JS∗
k
(λ)J
was used in [113,149] in the oscillation theory of discrete symplectic systems with general
nonlinear dependence on λ ∈ R.
The Lagrange identity established in (5.12) has many applications in the qualitative
theory of difference equations. Apart from the results in the following sections, it yields
for example the J-monotonicity of the fundamental matrix of system (Sλ). Following the
terminology from [114, pg. 7], a matrix M ∈ C2n×2n is called J-nondecreasing if iM∗JM ≥ iJ,
and M is J-nonincreasing if iM∗JM ≤ iJ. Similarly we define the corresponding notions
of a J-increasing or J-decreasing matrix. These concepts were used in [114] to study the
stability zones for periodic linear Hamiltonian differential systems. In a similar way, such
stability zones were studied in [129, 130] for the linear Hamiltonian difference systems
given in (2.6) with Hk ≡ 0 and in [49] for system (Sλ) with S[0]
k
= I.
Corollary 5.1.8. Let λ ∈ CS be fixed, Ψk(λ) ≥ 0 on [0, ∞)Z, and (λ) ∈ C([0, ∞)Z)2n×2n be
a fundamental matrix of system (Sλ) such that the matrix 0(λ) is symplectic, i.e., it satisfies
∗
0
(λ)J0(λ) = J. Then for every k ∈ [0, ∞)Z the matrix k(λ) is J-nondecreasing when
im(λ) > 0, or J-nonincreasing when im(λ) < 0. If, in addition, there exists N ∈ [0, ∞)Z such
that every nontrivial solution z(λ) ∈ C([0, ∞)Z)2n of system (Sλ) satisfies
N∑
k=0
z∗
k+1(λ)Ψk+1(λ)zk(λ) > 0, (5.15)
then the J-monotonicity of k(λ) is strict for all k ∈ [N + 1, ∞)Z, i.e., k(λ) is J-increasing when
im(λ) > 0, or J-decreasing when im(λ) < 0.
– 69 –
Chapter 5. Polynomial and analytic dependence on spectral parameter
Proof. By applying (5.12) to the fundamental matrix (λ) we get
i∗
k(λ)Jk(λ) − iJ = 2 im(λ)
k−1∑
j=0
∗
j+1(λ)Ψj(λ)j+1(λ). (5.16)
Since Ψk(λ) ≥ 0 for all k ∈ [0, ∞)Z, the sum on the right-hand side of (5.16) is zero for k = 0
and nonnegative for k ∈ [1, ∞)Z, so that k(λ) is J-nondecreasing when im(λ) > 0, and
it is J-nonincreasing when im(λ) < 0. Moreover, the additional assumption concerning
inequality (5.15) guarantees that the sum in (5.16) is positive definite for all k ∈ [N+1, ∞)Z,
so that k(λ) is J-increasing when im(λ) > 0, and it is J-decreasing when im(λ) < 0. ■
5.2 Special examples
In this section we show the connection of the generalized Lagrange identity from Theorem
5.1.6 with several special cases known in the literature. We also demonstrate that
a nonsingular weight matrix Ψk(λ) can be obtained when Sk(λ) is quadratic in λ, compare
with the weight matrix k defined in (2.1).
Example 5.2.1. The simplest example of system (Sλ) provides system (Sλ) with general
linear dependence on the spectral parameter studied in Chapter 2. Indeed, if the matrix
Sk(λ) is linear in λ, i.e., Sk(λ) = S[0]
k
+ λS[1]
k
and S
[j]
k
:= 0 for j = 2, 3, . . . , then Lemma 5.1.1
implies that
S[0]∗
k
JS[1]
k
= J, S[0]∗
k
JS[1]
k
+ S[1]∗
k
JS[0]
k
= 0, and S[1]∗
k
JS[1]
k
= 0.
In other words, the matrices Sk := S[0]
k
and Vk = S[1]
k
satisfy the first three conditions in (2.1)
and system (Sλ) reduces to (Sλ) with 5k(λ) := Sk(λ). In this case ε = ∞, i.e., CS = C, and
Λk(λ, ¯ν) = S[1]
k
JS[0]∗
k
. Hence Ψk(λ) = JS[1]
k
JS[0]∗
k
J, i.e., Ψk(λ) = k as defined in (2.1), which
shows that Theorem 5.1.6 generalizes Theorem 2.1.7 and Lemma 2.1.5. Consequently,
system (Sλ) includes all special cases of system (Sλ) mentioned in the introduction of
Chapter 2, see (2.7)–(2.10). ▲
Example 5.2.2. Now, let the dependence on λ in system (Sλ) be quadratic, i.e., S
[j]
k
≡ 0 for
j = 3, 4, . . . and
zk+1(λ) =
[
S[0]
k
+ λS[1]
k
+ λ2
S[2]
k
]
zk(λ), (5.17)
where S[0]
k
satisfies the first identity in (5.3), the matrices S[0]∗
k
JS[1]
k
and S[1]∗
k
JS[2]
k
are Hermitian,
S[2]∗
k
JS[2]
k
= 0, and
S[0]∗
k
JS[2]
k
+ S[1]∗
k
JS[1]
k
+ S[2]∗
k
JS[0]
k
= 0.
These conditions represent identity (5.3) with m = 0, . . . , 4, while for m = 5, 6, . . . the
sum in identity (5.3) is trivial. In particular, we consider system (5.17) with the special
quadratic dependence on λ given by
S[0]
k
=
(
Ak Bk
Ck Dk
)
, S[1]
k
=
(
0 Ak W[2]
k
−W[1]
k
Ak Ck W[2]
k
− W[1]
k
Bk
)
, S[2]
k
=
(
0 0
0 −W[1]
k
Ak W[2]
k
)
, (5.18)
where W[1]
k
and W[2]
k
are Hermitian n×n matrices, see also system (Qλ) in Section 5.4. Note
that the coefficients in (5.18) corresponds to (2.7) when W[2]
k
≡ 0. In this case the matrix
Sk(λ) can be factorized as
Sk(λ) =
(
I 0
−λW[1]
k
I
)
S[0]
k
(
I λW[2]
k
0 I
)
, (5.19)
– 70 –
5.2. Special examples
compare with (2.13). This shows that | det Sk(λ)| = det S[0]
k
(λ) = 1 on [0, ∞)Z for all λ ∈ C
as claimed in Theorem 5.1.3. If we put
Tk(λ) :=
(
0 Ak
−I Ck − λW[1]
k
Ak
)
and Wk := diag
{
W[1]
k
, W[2]
k
}
, (5.20)
then by (5.8) we get
Λ(λ, ¯ν) = S[1]
k
JS[0]∗
k
+ λS[2]
k
JS[0]∗
k
− ¯νS[0]
k
JS[2]∗
k
+ λ¯νS[2]
k
JS[1]∗
k
= −Tk(λ)Wk T ∗
k (ν).
Therefore by (5.13) we have
Ψk(λ) = −JTk(¯λ)Wk T ∗
k (¯λ)J and det Ψk(λ) = | det Ak |2
× det Wk, (5.21)
i.e., the matrix Ψk(λ) is no longer constant in λ and it is invertible if (and only if) the
matrices Ak, W[1]
k
, and W[2]
k
are invertible, compare with Ψk(λ) = k in the case of general
linear dependence on λ. However, the invertibility of the weight matrix Ψk(λ) can occur
only when system (5.17) with the coefficients specified in (5.18) corresponds to the linear
Hamiltonian difference system from (2.6) with Ak := I − A−1
k
, Bk := A−1
k
Bk, Ck := Ck A−1
k
,
Ek ≡ 0, Fk = W[2]
k
, and Gk = −W[1]
k
, see Remark 1.2.1(iv) and the identities in (1.29).
In addition, formula (5.4) yields that the multiplication of zk+1(λ) by T ∗
k
(¯λ)J produces
a backward shift in the second component. More precisely, if zk(λ) = (x∗
k
(λ), u∗
k
(λ))∗ solves
system (5.17) with (5.18), then by using the partially shifted notation
z[s]
k
(λ) := (x∗
k+1(λ), u∗
k(λ))∗
(5.22)
we obtain
T ∗
k (¯λ)Jzk+1(λ) = z[s]
k
(λ) and z∗
k+1(λ)JTk(¯λ) = −
[
T ∗
k (¯λ)Jzk+1(λ)
]∗
= −z[s]∗
k
(λ). (5.23)
Hence identity (5.9) can be written as
[
z∗
k(λ)Jzk(ν)
]
= −(¯λ−ν)z∗
k+1(λ)JTk(¯λ)Wk T ∗
k (¯ν)Jzk+1(ν) = (¯λ−ν)z[s]∗
k
(λ)Wk z[s]
k
(ν). (5.24)
Since the weight matrix Wk in (5.24) is independent of λ, we can associate with the system
in hand a semi-inner product and a semi-norm, which are independent of λ, see (5.41)
below. We note that the latter observation is crucial for the invariance of the limit circle
case derived in Section 5.4. ▲
In the following example we investigate the connection between the linear Hamiltonian
difference system from (2.6) and system (Sλ).
Example 5.2.3. According to Remark 1.2.1(iv), see identity (1.28), system (2.6) can be
written as system (Sλ) with the coefficient matrix Sk(λ) =
( Ak(λ) Bk(λ)
Ck(λ) Dk(λ)
)
, where
Ak(λ) := Ak(λ) = (I − Ak − λEk)−1
, Bk(λ) := Ak(λ)(Bk + λFk),
Ck(λ) := (Ck + λGk)Ak(λ), Dk(λ) := I − A∗
k − λE∗
k + (Ck + λGk)Ak(λ)(Bk + λFk).
}
(5.25)
We claim that the matrix Ak(λ) is polynomial in λ, and consequently the corresponding
system (Sλ) is polynomial in λ. Let us fix k ∈ [0, ∞)Z. By the definition of the determinant,
the function d(λ) := det(I −Ak −λEk) is a polynomial of degree at most n. The assumption
on the existence of Ak(λ) for all λ ∈ C then implies that d(λ) 0 for all λ ∈ C. Thus,
– 71 –
Chapter 5. Polynomial and analytic dependence on spectral parameter
d(λ) ≡ d 0 on C and the matrix I − Ak − λEk is unimodular. This yields that Ak(λ) is also
a polynomial matrix in λ (of degree at most n − 1) and hence by (5.25), the matrix Sk(λ) is
in this case polynomial of degree at most n + 1. Although we do not calculate the matrix
Λ(¯λ, ν) explicitly, the corresponding Lagrange identity can be written as in (5.24) with Wk
replaced by −JWk; cf. [36, Formula (2.55)]. Indeed, one easily observes that
Sk(λ)JS∗
k(ν) =
(
Ak(λ)B∗
k
(ν) − Bk(λ)A∗
k
(ν) Ak(λ)D∗
k
(ν) − Bk(λ)C∗
k
(ν)
Ck(λ)B∗
k
(ν) − Dk(λ)A∗
k
(ν) Ck(λ)D∗
k
(ν) − Dk(λ)C∗
k
(ν)
)
,
and similarly as in (5.23) we get
zk+1(λ) = −JT ∗−1
k (¯λ)z[s]
k
(λ), where Tk(λ) :=
(
0 Ak(λ)
−I Ck(λ)
)
.
Therefore, following the calculation in the proof of Theorem 5.1.6 we obtain
[
z∗
k(λ)Jzk(ν)
]
= z[s]∗
k
(λ)T −1
k (¯λ)[J − Sk(¯λ)JS∗
k(¯ν)]T ∗−1
k (¯ν)z[s]
k
(ν)
= −(¯λ − ν)z[s]∗
k
(λ)JWk z[s]
k
(ν). (5.26)
Especially, if Ek ≡ 0, then the matrix Ak(λ) ≡ Ak does not depend on λ and in this case
system (2.6) can be written as system (5.17) with the special quadratic dependence on λ
specified in (5.18) with Ak being invertible, see also [142, Formula (2.3) and Lemma 2.2]. ▲
In the last example of this section we consider system (Sλ) with the truly analytic (i.e.,
nonpolynomial) dependence on λ, which was studied in [48,49].
Example 5.2.4. Let S
[j]
k
:= (1/j!)E
j
k
for j = 0, 1, . . . , where Ek ∈ C2n×2n is Hamiltonian for
all k ∈ [0, ∞)Z, i.e., E∗
k
J + JEk = 0. Then CS = C and the coefficient matrix Sk(λ) is of the
exponential type, i.e.,
Sk(λ) =
∞∑
j=0
λj
j!
E
j
k
= exp(λEk). (5.27)
Then by (5.8), (1.9)–(1.10), and the Hamiltonian property of Ek we obtain
Λ(λ, ¯ν) =
∞∑
j=1
(−1)j (λ − ¯ν)j−1
j!
J(E∗
k)j
, (5.28)
compare with [49, pg. 6] or [48, Section 2]. The Lagrange identity has the same form as
in (5.9) with the corresponding Λk(¯λ, ν). Especially, let n = 1 and consider the matrix Sk(λ)
as in (5.27) with
Ek ≡ E :=
(
i 1
−1 i
)
.
Then by (1.9) we have det Sk(λ) = eλ tr E = e2iλ = e−2 im(λ) e2i re(λ) for all k ∈ [0, ∞)Z and any
λ ∈ C. Thus | det Sk(λ)| = e−2 im(λ) and it is equal to one if only if λ ∈ R, which agrees with
the symplecticity of the matrix Sk(λ) on the real line; compare with Theorem 5.1.3 and see
also Example 5.3.9. ▲
– 72 –
5.3. Weyl–Titchmarsh theory
5.3 Weyl–Titchmarsh theory
In this section we focus on an eigenvalue problem and the Weyl–Titchmarsh theory for
system (Sλ) with analytic or polynomial dependence on λ. We show that the main results
of Chapter 2 remain valid also for the latter system, when we modify the corresponding
Atkinson-type conditions to this more general setting. The solutions are weighted with
respect to the Hermitian matrix Ψk(λ) defined in (5.13), that is,
||z||Ψ(λ) :=
√
⟨z, z⟩Ψ(λ) and ⟨z, ˜z⟩Ψ(λ) :=
∞∑
k=0
z∗
k+1 Ψk(λ) ˜zk+1. (5.29)
The expression of the bilinear form in (5.29) justifies the restriction of λ ∈ CS to those
values for which Ψk(λ) ≥ 0. Since this condition can be violated for some λ ∈ CS, we
denote by CΨ and CΨ,N the subsets of CS such that
CΨ,N :=
{
λ ∈ CS | Ψk(λ) ≥ 0 for all k ∈ [0, N]Z
}
,
CΨ :=
{
λ ∈ CS | Ψk(λ) ≥ 0 for all k ∈ [0, ∞)Z
}
,
where N ∈ [0, ∞)Z. An example with CΨ ⊊ CS can be found in [A20, Example 5.7] for
a continuous analogue of system (Sλ). Our treatment is based on the Lagrange identity
derived in Theorem 5.1.6 and a construction of the Weyl disks. Nevertheless, the proofs
are basically the same as for the linear dependence on λ in Chapter 2 and hence they are
omitted. For brevity, we do not keep the precise identification of the minimal assumptions
as in Chapter 2 and slightly simplify some formulations.
Throughout this section, let α ∈ be given, see (2.19), and Z(λ), Z(λ) ∈ C([0, ∞)Z)2n×n
be the two components of the fundamental matrix k(λ) =
(
Zk(λ), Zk(λ)
)
of system (Sλ)
satisfying 0(λ) = (α∗, −Jα∗), i.e., the solutions Z(λ) and Z(λ) are determined by the initial
conditions Z0(λ) = α∗ and Z0(λ) = −Jα∗, compare with (2.22). The fundamental matrix
(λ) then satisfies the identities
∗
k(¯λ)Jk(λ) = J and k(λ)J∗
k(¯λ) = J for all k ∈ [0, ∞)Z,
see Theorem 5.1.6 and compare with Lemma 2.1.6.
Now, let us fix N ∈ [0, ∞)Z and β ∈ . If we associate with system (Sλ) the following
eigenvalue problem
(Sλ), k ∈ [0, N]Z, λ ∈ CS, αz0(λ) = 0, βzN+1(λ) = 0, (5.30)
then it follows as in Theorem 2.2.3 that the eigenvalues of problem (5.30) are characterized
by det βZN+1(λ) = 0 and the corresponding eigenfunctions are of the form Z(λ)d with
a nonzero d ∈ Ker βZN+1(λ). Moreover, we introduce the following hypothesis, compare
with Hypothesis 2.2.2.
Hypothesis 5.3.1 (Weak Atkinson condition – finite). For any λ ∈ CΨ,N KR every column
z(λ) of the solution Z(λ) satisfies
N∑
k=0
z∗
k+1(λ)Ψk(λ)zk+1(λ) > 0.
Then, under Hypothesis 5.3.1, all eigenvalues of problem (5.30) restricted to the set
CΨ,N are real and eigenfunctions corresponding to different eigenvalues are orthogonal
with respect to the semi-inner product ⟨·, ·⟩Ψ(λ),N defined similarly as in (5.29) with the
sum over the finite discrete interval [0, N]Z.
– 73 –
Chapter 5. Polynomial and analytic dependence on spectral parameter
The M(λ)-function for system (Sλ) is defined in the same way as in Definition 2.2.4,
i.e., Mk(λ) := −[βZk(λ)]−1βZk(λ), and it satisfies the properties established in Lemma 2.2.5
and Theorem 2.2.7. In particular, M∗
k
(λ) = Mk(¯λ) and Mk(λ) is analytic in λ.
For the rest of this section we consider system (Sλ) on [0, ∞)Z and study its square
summable solutions. For this purpose we will need the following condition.
Hypothesis 5.3.2 (Weak Atkinson condition – infinite). For λ ∈ CSKR such that λ, ¯λ ∈ CΨ
there exists N4 ∈ [0, ∞)Z such that for ν ∈ {λ, ¯λ} we have
N4∑
k=0
z∗
k+1(ν)Ψk(ν)zk+1(ν) > 0 (5.31)
for every column z(ν) of the solution Z(ν) of system (Sν).
We note that the number N4 in Hypothesis 5.3.2 depends in general on the chosen λ
(and ¯λ). This is a weaker condition than in Hypothesis 2.3.7, where it was considered for
all λ ∈ C. The results of this section are phrased in terms of the following set
CA :=
{
λ ∈ CSKR | Hypothesis 5.3.2 holds at λ
}
,
which is associated with the above Atkinson-type condition. Then, by definition, we have
λ ∈ CA if and only if ¯λ ∈ CA, i.e., the set CA is symmetric with respect to the real axis. This
observation is also very important for the development of the present theory.
For M ∈ Cn×n we define the Weyl solution X(λ, M) ∈ C([0, ∞)Z)2n×n of system (Sλ) by
Xk(λ, M) := k(λ)(I, M∗
)∗
,
where (λ) is the fundamental matrix of system (Sλ) specified above, cf. (2.23). Moreover,
we utilize the Hermitian matrix-valued function E : [0, ∞)Z × CA × Cn×n → Cn×n given by
Ek(λ, M) := iδ(λ)X∗
k (λ, M)JXk(λ, M),
compare with (2.32). This function is used for the definition of the Weyl disk Dk(λ) and
the Weyl circle Ck(λ), i.e.,
Dk(λ) :=
{
M ∈ Cn×n
| Ek(λ, M) ≤ 0
}
, Ck(λ) :=
{
M ∈ Cn×n
| Ek(λ, M) = 0
}
.
Since for k = 0 we have E0(λ, M) = −2δ(λ) im(M), it follows from (5.12) that for λ ∈ CA
and k ∈ [1, ∞)Z the elements of Dk(λ) are characterized by the inequality
k−1∑
j=0
X∗
j+1(λ, M)Ψj(λ)Xj+1(λ, M) ≤
im(M)
im(λ)
, (5.32)
compare with Theorem 2.3.5. Similarly, the elements of Ck(λ) are characterized by the
equality in (5.32). The following geometric description of the Weyl disk and the Weyl
circle can be derived as in Section 2.3. If we set
Gk(λ) := iδ(λ)Z
∗
k(λ)JZk(λ), Hk(λ) := iδ(λ)Z
∗
k(λ)JZk(λ), (5.33)
then Hk(λ) is Hermitian, H0(λ) = 0, and identity (5.12) yields
Hk(λ) = 2| im(λ)|
k−1∑
j=0
Z
∗
j+1(λ)Ψj(λ)Z j+1(λ) ≥ 0. (5.34)
This shows that Hk(λ) is nondecreasing in k ∈ [0, ∞)Z. Moreover, the symmetry of the set
CA with respect to the real axis and Hypothesis 5.3.2 guarantee that the matrices Hk(λ)
and Hk(¯λ) are positive definite for k ∈ [N4 + 1, ∞)Z. We summarize the main properties of
the Weyl disks in the following theorem.
– 74 –
5.3. Weyl–Titchmarsh theory
Theorem 5.3.3. Let α ∈ , λ ∈ CA, and suppose that Hypothesis 5.3.2 holds. Then for all
k ∈ [N4 + 1, ∞)Z the Weyl disk and the Weyl circle admit the representations
Dk(λ) =
{
Pk(λ) + Rk(λ)VRk(¯λ) | V ∈ V
}
, Ck(λ) =
{
Pk(λ) + Rk(λ)URk(¯λ) | U ∈ U
}
, (5.35)
where the center Pk(λ) and the matrix radii Rk(λ), Rk(¯λ) are defined by
Pk(λ) := −H−1
k (λ)Gk(λ), Rk(λ) := H−1/2
k
(λ), Rk(¯λ) := H−1/2
k
(¯λ), (5.36)
and U, V are the sets defined in (2.39). Moreover, the Weyl disk Dk(λ) is closed, convex, and
Dk(λ) ⊆ Dj(λ) for all k, j ∈ [N4 + 1, ∞)Z with k ≥ j.
Proof. The proof follows the same arguments as in Theorem 2.3.8. In particular, for the
representations given in (5.35) we utilize the identity
Ek(λ, M) = [H−1
k (λ)Gk(λ) + M]∗
Hk(λ)[H−1
k (λ)Gk(λ) + M] − H−1
k (¯λ) (5.37)
for k ∈ [N4 + 1, ∞)Z, which is obtained by completing Ek(λ, M) to a square, see formulas
(2.38) and (2.43). Expression (5.37) uses the invertibility of Hk(λ) and Hk(¯λ) for
k ∈ [N4 + 1, ∞)Z, which is guaranteed by the assumption λ ∈ CA. In fact, the motivation
for the complicated form of Hypothesis 5.3.2 comes from the above symmetry argument
with respect to λ and ¯λ. ■
The latter properties of the Weyl disks imply that the intersection of all Dk(λ) for
k ∈ [N4 + 1, ∞)Z is nonempty, closed, and convex. Thus we define the limiting Weyl disk
D+(λ) := lim
k→∞
Dk(λ) =
∩
k∈[N4+1,∞)Z
Dk(λ).
From Theorem 5.3.3 and inequality (5.32) we obtain the following result. It extends
Corollaries 2.3.11 and 2.3.12 to the case of the analytic dependence on λ.
Theorem 5.3.4. Let α ∈ , λ ∈ CA, and suppose that Hypothesis 5.3.2 holds. Then
D+(λ) =
{
P+(λ) + R+(λ)VR+(¯λ) | V ∈ V
}
, (5.38)
where the limiting center P+(λ) and the limiting matrix radii R+(λ), R+(¯λ) are given by
P+(λ) := lim
k→∞
Pk(λ), R+(λ) := lim
k→∞
Rk(λ) ≥ 0, R+(¯λ) := lim
k→∞
Rk(¯λ) ≥ 0. (5.39)
In addition, a matrix M ∈ Cn×n belongs to the limiting Weyl disk D+(λ) if and only if
∞∑
k=0
X∗
k+1(λ, M)Ψk(λ)Xk+1(λ, M) ≤
im(M)
im(λ)
. (5.40)
We note that by (5.34) and (5.36) the limit of Rk(λ) as k → ∞ exist and is positive
semidefinite, while the proof of the existence of the limit of the matrices Pk(λ) is based
on the fixed point argument as in Theorem 2.3.9. The statement of Theorem 5.3.4 follows
from Theorem 5.3.3 and formula (5.32).
Let λ ∈ CΨ be fixed. We now turn our attention to the number of linearly independent
square summable solutions of system (Sλ). By ℓ2
Ψ(λ)
we denote the space of all sequence
on [0, ∞)Z, which are square summable with respect to the weight Ψ(λ), i.e.,
ℓ2
Ψ(λ) :=
{
z ∈ C([0, ∞)Z)2n
| ||z||Ψ(λ) < ∞
}
,
– 75 –
Chapter 5. Polynomial and analytic dependence on spectral parameter
where the semi-norm ||·||Ψ(λ) is defined in (5.29) and Ψk(λ) ≥ 0 on [0, ∞)Z. The space ℓ2
Ψ(λ)
generally depends on the value of λ, but in some special cases it may be independent
of λ, see Example 5.2.1 with Ψk(λ) ≡ k. Furthermore, in view of (5.24) or (5.26) in
Examples 5.2.2 and 5.2.3, i.e., we may consider for system (Sλ) with the coefficients
specified in (5.18) or (5.25) the space
ℓ2
W :=
{
z ∈ C([0, ∞)Z)2n
∞∑
k=0
z[s]∗
k
Wk z[s]
k
< ∞
}
, (5.41)
which does not depend on λ, see also Section 5.4.
We are interested in the subspace N(λ) ⊆ ℓ2
Ψ(λ)
consisting of all square summable
solutions of system (Sλ), i.e.,
N(λ) :=
{
z ∈ ℓ2
Ψ(λ) | z solves system (Sλ)
}
. (5.42)
If λ ∈ CA, then by inequality (5.40)the columns of the Weylsolution X(λ, M)corresponding
to the matrices M ∈ D+(λ) are linearly independent and all belong to N(λ). This means that
n ≤ dim N(λ) ≤ 2n for all λ ∈ CA, which justifies the classification of system (Sλ) as being
in the limit point case if dim N(λ) = n, and as being in the limit circle case if dim N(λ) = 2n.
The remaining cases with n + 1 ≤ dim N(λ) ≤ 2n − 1 are called intermediate. Moreover,
one can verify that the results of Theorem 2.4.1–Corollary 2.4.10 in Section 2.4 hold with
exactly the same proofs also in the case of the analytic dependence on λ. Especially, the
following extension of Theorem 2.4.8 to the analytic dependence on λ is true.
Theorem 5.3.5. Let α ∈ , λ ∈ CA, and suppose that Hypothesis 5.3.2 holds. Then system (Sλ)
has exactly n + rank R+(λ) linearly independent square summable solutions, i.e.,
dim N(λ) = n + rank R+(λ),
where R+(λ) is the matrix radius of the limiting Weyl disk D+(λ) defined in (5.39).
By combining Theorem 5.3.5 and identity (5.38) we obtain the following limit point
and limit circle classification of system (Sλ) in terms of the rank of R+(λ), compare with
Theorem 2.4.3 and Corollary 2.4.10.
Corollary 5.3.6. Let α ∈ , λ ∈ CA, and Hypothesis 5.3.2 hold. Then system (Sλ) is
(i) in the limit point case if and only if R+(λ) = 0, in which case D+(λ) = {P+(λ)} and
D+(¯λ) = {P+(¯λ)},
(ii) in the limit circle case if and only if R+(λ) is invertible.
Remark 5.3.7. We note that the results of Chapter 3 regarding the Weyl–Titchmarsh theory
for discrete symplectic systems with jointly varying endpoints hold in the same way for
system with the analytic dependence on λ under the appropriate strong Atkinson-type
conditions including all nontrivial solutions, see Hypotheses 3.1.1 and Hypothesis 2.4.11.
Finally, we illustrate the results of the Weyl–Titchmarsh theory for system (Sλ) by two
interesting examples with the exponential dependence on λ discussed in Example 5.2.4.
Example 5.3.8. In this example we show that the discrete symplectic system
zk+1(λ) = exp(λJ)zk(λ). (5.43)
is in the limit point case for every λ ∈ CKR and we calculate the unique 2n × n solution
(up to an invertible multiple) of system (5.43) whose columns lie in ℓ2
Ψ(λ)
and form a basis
– 76 –
5.3. Weyl–Titchmarsh theory
of N(λ), i.e., the Weyl solution X(λ, P+(λ)). System (5.43) corresponds to the system from
Example 5.2.4 with Ek := E ≡ J, which satisfies the condition E∗
k
J + JEk = 0. Moreover,
we have Sk(λ) = exp(λJ) = (cos λ)I + (sin λ)J and CS = C, see also [16, Example 11.3.4].
For simplicity we perform the calculations below in the scalar case, i.e., for n = 1. The
general case follows with the same arguments upon multiplication by the n × n or 2n × 2n
identity matrices at appropriate places. If we choose α = (1, 0), then the fundamental
matrix k(λ) of system (5.43) with 0(λ) = I is given by
k(λ) = exp(kλJ) = (cos kλ)I + (sin kλ)J =
(
cos kλ sin kλ
− sin kλ cos kλ
)
, k ∈ [0, ∞)Z,
and so Zk(λ) = (sin kλ, cos kλ)⊤. Since the powers of J repeat in a cycle of length four, we
obtain for any k ∈ [0, ∞)Z by (5.28) that Λk(¯λ, λ) = −I for all λ ∈ R, while for λ ∈ CKR we
calculate (with p := im(λ) 0)
Λk(¯λ, λ) =
∞∑
j=1
(−2ip)j−1
j!
Jj+1
=
1
2ip
∞∑
j=1
(−1)j+1 (2ip)2j
(2j)!
J +
1
2ip
∞∑
j=0
(−1)j+1 (2ip)2j+1
(2j + 1)!
I
=
cosh 2p − 1
2p
iJ −
sinh 2p
2p
I =
sinh p
p
[
(sinh p)iJ − (cosh p)I
]
,
where we used the well-known identities sinh 2x = 2 sinh x cosh x, cosh 2x = 2 sinh2
x + 1,
i sinh p = sin(ix), and cosh p = cos(ip). Hence by (5.13) we have
Ψk(λ) ≡ Ψ(λ) =



sinh p
p


cosh p −i sinh p
i sinh p cosh p

 > 0 for all λ ∈ CKR,
I for all λ ∈ R,
(5.44)
see also (5.14). Thus CΨ = C. By the definitions of Hk(λ) and Gk(λ) in (5.33) we get
Hk(λ) = iδ(λ)(sin k¯λ cos kλ − cos k¯λ sin kλ) = δ(λ) sinh(2k im(λ)),
Gk(λ) = −iδ(λ)(sin k¯λ sin kλ + cos k¯λ cos kλ) = −iδ(λ) cosh(2k im(λ)).
Note that the same value for Hk(λ) is of course obtained from formula (5.34) after some
calculations. This shows that Hypothesis 5.3.2 is satisfied for any λ ∈ CKR and any
N4 ∈ [0, ∞)Z, i.e., CA = CKR. The relations in (5.36) yield
Pk(λ) = i coth(2k im(λ)) and Rk(λ) = 1/
√
sinh(2k| im(λ)|) for all k ∈ [1, ∞)Z.
The center and radius of the limiting disk D+(λ) are then
P+(λ) = lim
k→∞
Pk(λ) = iδ(λ) and R+(λ) = lim
k→∞
Rk(λ) = 0,
which shows that system (5.43) is in the limit point case for every λ ∈ CKR by Corollary
5.3.6(i). Moreover, the space N(λ) of square summable solutions of system (5.43)
with λ ∈ CKR is generated by the Weyl solution
Xk(λ, P+(λ)) = k(λ)
(
1
P+(λ)
)
=
(
cos kλ + iδ(λ) sin kλ
− sin kλ + iδ(λ) cos kλ
)
=
(
1
iδ(λ)
)
eiδ(λ)kλ
,
– 77 –
Chapter 5. Polynomial and analytic dependence on spectral parameter
for which (we again substitute p := im(λ)) we have
X(λ, P+(λ))
2
Ψ(λ)
=
∞∑
k=0
X∗
k+1(λ, P+(λ))Ψk(λ)Xk+1(λ, P+(λ))
=
2 sinh p
p
× [cosh p + δ(λ) sinh p] ×
∞∑
k=0
e−2|p|(k+1)
=
2 sinh p
p
× [cosh p + δ(λ) sinh p] ×
e−2|p|
1 − e−2|p|
=
1
|p|
.
This shows that ||X(λ, P+(λ))||Ψ(λ) = 1/
√
| im(λ)| < ∞, and so indeed X(λ, P+(λ)) ∈ ℓ2
Ψ(λ)
for every λ ∈ CKR. On the other hand, we also have
Z(λ)
2
Ψ(λ)
=
∞∑
k=0
Z
∗
k+1(λ)Ψk(λ)Zk+1(λ)
(5.34)
=
1
2| im(λ)|
lim
k→∞
Hk(λ)
=
1
2| im(λ)|
lim
k→∞
sinh(2k| im(λ)|) = ∞,
i.e., Z(λ) ℓ2
Ψ(λ)
. Thus, again we get that dim N(λ) = 1 for any λ ∈ CKR, see also the
proof of Theorem 2.4.3. Similarly, in arbitrary dimension n we get that the n columns of
the Weyl solution X(λ, P+(λ)) are linearly independent and they belong to ℓ2
Ψ(λ)
, while
the n columns of Z(λ) are linearly independent and they do not belong to ℓ2
Ψ(λ)
. Hence,
dim N(λ) = n and system (5.43) is in the limit point case for all λ ∈ CKR. ▲
Now we give a counterexample for the invariance of the limit circle case when the
dependence on λ is analytic and nonpolynomial.
Example 5.3.9. Let us consider system (Sλ) with the coefficient matrix Sk(λ) from Example
5.2.4 with Ek := E ≡ iI + J, i.e., the system
zk+1(λ) = exp(λE)zk(λ), exp(λE) = eiλ
[(cos λ)I + (sin λ)J], (5.45)
see again [16, Example 11.3.4]. Then CS = C and the fundamental matrix k(λ) of
system (5.45) corresponding to the choice α = (1, 0) is given by
k(λ) = exp(kλE) = eikλ
[(cos kλ)I + (sin kλ)J], k ∈ [0, ∞)Z.
Since (E∗)j = −(−2i)j−1 E for j ≥ 1, it follows by (5.28) that for any k ∈ [0, ∞)Z we have
Λk(¯λ, λ) = iE if λ ∈ R and Λk(¯λ, λ) = i
4p (e4p − 1)E if λ ∈ CKR with p := im(λ). Hence
identity (5.13) yields
Ψk(λ) ≡ Ψ(λ) =



−
i(e4 im(λ) − 1)
4 im(λ)
E for all λ ∈ CKR,
−iE for all λ ∈ R.
(5.46)
Moreover, iE ≤ 0, which yields through (5.46) that Ψ(λ) ≥ 0 for all λ ∈ C, i.e., CΨ = C.
Note that in this case Ψ(λ) is singular for all λ ∈ C, compare with (5.44).
The left-hand side of (5.31) with zk(λ) = Zk(λ) = eikλ(sin kλ, cos kλ)⊤ has the form
N4∑
k=0
z∗
k+1(λ)Ψ(λ)zk+1(λ) =



1 − e−4(N4+1) im(λ)
4 im(λ)
, λ ∈ CKR,
N4 + 1, λ ∈ R.
– 78 –
5.4. Special quadratic dependence and limit circle case
Therefore the inequality in (5.31) is satisfied for all λ ∈ C and any N4 ∈ [0, ∞)Z, which
implies CA = CKR. By (5.33), (5.34), and (5.36) we get
Hk(λ) = δ(λ)
(
1 − e−4k im(λ)
)
/2, Gk(λ) = −iδ(λ)e−2k im(λ)
cosh(2k im(λ)),
Pk(λ) = i coth(2k im(λ)), R2
k(λ) =
2
δ(λ)
(
1 − e−4k im(λ)
).
Thus by (5.39) we have P+(λ) = iδ(λ) for every λ ∈ CA and R+(λ) =
√
2 for λ ∈ C+, while
R+(λ) = 0 for λ ∈ C−. This shows by Corollary 5.3.6 that system (5.45) is in the limit circle
case for λ ∈ C+ and in the limit point case for λ ∈ C−. The space N(λ) is then generated
by the columns of the fundamental matrix (λ) when λ ∈ C+, and by the Weyl solution
Xk(λ, P+(λ)) ≡ (1, −i)⊤ with ||X(λ, −i)||Ψ(λ) = 0 when λ ∈ C−. For completeness we note
that system (5.45) is in the limit point case also for all λ ∈ R with X(λ, −i) being the unique
square summable solution (up to a constant multiple). ▲
5.4 Special quadratic dependence and limit circle case
In Example 5.3.9 it was shown that the dimension of N(λ) may vary with respect to λ,
even when Hypothesis 5.3.2 is satisfied. In particular, system (Sλ) can be in the limit circle
case for some value λ and in the limit point case for another one. This situation is not
possible when the dependence on λ is only linear as we derived in Chapter 4 and stated
in Theorem 2.4.17. In this section we prove a similar invariance of the limit circle case for
system (Sλ) with the special quadratic dependence on λ from Example 5.2.2. In this case
we can choose the associated space of square summable solutions to be independent of λ,
which is a key ingredient for this result. Note that this property was trivially satisfied
also in the previous chapters.
5.4.1 Results for one system
Let us consider system (5.17) with the coefficients specified in (5.18) or equivalently
(suppressing the argument λ)
xk+1 = Ak xk +
(
Bk + λAW[2]
k
)
uk, uk+1 = Ck xk +
(
Dk + λCk W[2]
k
)
uk − λW[1]
k
xk+1, (Qλ)
where A, B, C, D, W[1]
, W[2]
∈ C([0, ∞)Z)n×n are such that the matrix S[0]
k
in (5.18) satisfies
the first equality in (5.3), W[1]
k
and W[2]
k
are Hermitian, and
Wk := diag
{
W[1]
k
, W[2]
k
}
≥ 0 for all k ∈ [0, ∞)Z. (5.47)
Recalling the notation from (5.22), we associate with system (Qλ) the space of all square
summable sequences with respect to the weight matrix Wk defined in (5.41), i.e., ℓ2
W
.
Since from (5.21) and (5.24) one infers that z∗
k+1
(λ)Ψk(λ)zk+1(λ) = z[s]∗
k
(λ)Wk z[s]
k
(λ) for
all solutions of system (Qλ), the space of all solutions being in ℓ2
W
is the same as the
corresponding space N(λ) defined in (5.42). This means that we have
N(λ) =
{
z ∈ ℓ2
W | z solves system (Qλ)
}
and system (Qλ) is in the limit point case when dim N(λ) = n and in the limit circle case
when dim N(λ) = 2n.
– 79 –
Chapter 5. Polynomial and analytic dependence on spectral parameter
The following result concerning the invariance of the limit circle case for system (Qλ)
generalizes [142, Theorem 5.5] for system (2.6) with Ek ≡ 0, which corresponds to system
(Qλ) with Ak being invertible on [0, ∞)Z. The proof is given in Subsection 5.4.2 below,
where we establish a more general statement (Theorem 5.4.5) for two systems of the
form (Qλ) as in Chapter 4.
Theorem 5.4.1. Let (5.47) hold. If there exists λ0 ∈ C such that system (Qλ0
) is in the limit circle
case, then system (Qλ) is in the limit circle case for every λ ∈ C.
From Theorem 5.4.1 we obtain the following simple criterion for the limit circle case
in terms of the norms of the coefficients S[0]
k
and Wk; cf. Corollary 2.4.19. It extends the
statement in [134, Theorem 6.3] for system (2.6) with Ek ≡ 0.
Corollary 5.4.2. Let (5.47) hold and assume that
∞∑
k=0
||S[0]
k
− I||1 < ∞ and
∞∑
k=0
||Wk ||1 < ∞. (5.48)
Then system (Qλ) is in the limit circle case for all λ ∈ C.
Proof. The conditions in (5.48) imply that system (Q0) is in the limit circle case. Therefore
the result follows from Theorem 5.4.1. Alternatively, this statement can be derived as
a special case of Corollary 5.4.6 below regarding two systems. ■
In the scalar case we get from Theorems 5.4.1 and 5.3.5 the following limit point
criterion for system (Qλ).
Theorem 5.4.3. Let n = 1 and assume that condition (5.47) and Hypothesis 5.3.2 hold. If there
exists λ0 ∈ C such that system (Qλ0
) is in the limit point case, then system (Qλ) is in the limit
point case for every λ ∈ CA.
Proof. Assume that for some λ1 ∈ CA system (Qλ1
) is not in the limit point case. Then by
n = 1 and Theorem 5.3.5 we know that (Qλ1
) is in the limit circle case. Consequently, by
Theorem 5.4.1, system (Qλ) is in the limit circle case for all λ ∈ C, which contradicts the
original assumption that system (Qλ0
) is in the limit point case. ■
By combining Theorems 5.4.1 and 5.4.3 we obtain the following extension of the Weyl
alternative, i.e., the dichotomy between the limit point and limit circle classifications of
system (Qλ) for all suitable λ; cf. Corollary 2.4.23.
Corollary 5.4.4 (Weyl alternative). Let n = 1 and assume that condition (5.47) and Hypothesis
5.3.2 hold. Then system (Qλ) is either in the limit circle case for all λ ∈ C, or in the limit point
case for all λ ∈ CA.
5.4.2 Results for two systems
Motivated by the results in Chapter 4, we consider instead of system (Qλ) two systems of
the same form (suppressing the argument λ)
ˆxk+1 = Ak ˆxk +
(
Bk + λAW[2]
k
)
ˆuk, ˆuk+1 = Ck ˆxk +
(
Dk + λCk W[2]
k
)
ˆuk − λW[1]
k
ˆxk+1, (Qλ)
˜xk+1 = Ak ˜xk +
(
Bk + λAW[2]
k
)
˜uk, ˜uk+1 = Ck ˜xk +
(
Dk + λCk W[2]
k
)
˜uk − λW[1]
k
˜xk+1, (Qλ)
where A, B, C, Dk, A, B, C, Dk, W[1]
, W[2]
∈ C([0, ∞)Z)n×n with W[1]
k
and W[2]
k
being Hermitian
and satisfying (5.47). Note that the weight matrices W[1]
k
and W[2]
k
are the same in both
– 80 –
5.4. Special quadratic dependence and limit circle case
systems (Qλ) and (Qλ). This is justified by the requirement of having the same space
of square summable functions associated with these systems. The coefficient matrices
S
[0]
, S
[1]
, S
[2]
∈ C([0, ∞)Z)2n×2n and S
[0]
, S
[1]
, S
[2]
C([0, ∞)Z)2n×2n of systems (Qλ) and (Qλ),
respectively, are defined analogously to (5.18). We assume that systems (Qλ) and (Qλ) are
in general non-symplectic, i.e., we do not impose that the matrices S
[0]
k and S
[0]
k are such
that the first equality in (5.3) holds. Instead we assume on [0, ∞)Z the combined identity
S
[0]∗
k JS
[0]
k = J. (5.49)
By using the block structure of S
[0]
k and S
[0]
k we obtain that equality (5.49) is equivalent to
A
∗
k Ck = C
∗
k Ak, D
∗
k Bk = B
∗
k Dk, A
∗
k Dk − C
∗
k Bk = I and D
∗
k Ak − B
∗
kCk = I. (5.50)
Consequently, the coefficient matrices also satisfy the following identities
S
[0]∗
k JS
[1]
k + S
[1]∗
k JS
[0]
k = 0, (5.51)
S
[0]∗
k JS
[2]
k + S
[1]∗
k JS
[1]
k + S
[2]∗
k JS
[0]
k = 0, (5.52)
S
[1]∗
k JS
[2]
k + S
[2]∗
k JS
[1]
k = 0, S
[2]∗
k JS
[2]
k = 0. (5.53)
In addition, identity (5.49) is equivalent to
S
[0]
k JS
[0]∗
k = J, (5.54)
which can be written in terms of the n × n blocks as
Ak B
∗
k = Bk A
∗
k, Dk C
∗
k = Ck D
∗
k, Ak D
∗
k − Bk C
∗
k = I and Dk A
∗
k − CkB
∗
k = I. (5.55)
Since identity (5.49) is equivalent also with S
[0]∗
k JS
[0]
k = J and S
[0]
k JS
[0]∗
k = J, the block
matrices satisfy the relations
A
∗
k Ck = C
∗
k Ak, D
∗
k Bk = B
∗
k Dk, A
∗
k Dk − C
∗
k Bk = I, D
∗
k Ak − B
∗
kCk = I, (5.56)
Ak B
∗
k = Bk A
∗
k, Dk C
∗
k = Ck D
∗
k, Ak D
∗
k − Bk C
∗
k = I, Dk A
∗
k − CkB
∗
k = I. (5.57)
Note that (5.49) or (5.54) trivially implies that det S
[0]∗
k × det S
[0]
k = 1.
Let Sk(λ) := S
[0]
k +λS
[1]
k +λ2 S
[2]
k and Sk(λ) := S
[0]
k +λS
[1]
k +λ2 S
[2]
k be the coefficient matrices
of systems (Qλ) and (Qλ), respectively. Then
S
∗
k(λ)JSk(¯λ) = J and Sk(λ)JS
∗
k(¯λ) = J with S
−1
k (λ) = −JS
∗
k(¯λ)J, (5.58)
which means that Sk(λ) and Sk(λ) are invertible for all k ∈ [0, ∞)Z and any λ ∈ C. Therefore
any initial value problems associated with systems (Qλ) and (Qλ) posses unique solutions
on [0, ∞)Z for any initial value given at any point in [0, ∞)Z. Moreover, the fundamental
matrices of systems (Qλ) and (Qλ) are in this case invertible on the discrete interval [0, ∞)Z.
Since by (5.19) we have
det Sk(λ) = det S
[0]
k and det Sk(λ) = det S
[0]
k , (5.59)
it follows that
det S
∗
k(λ) × det Sk(λ) = 1 for all k ∈ [0, ∞)Z and any λ ∈ C. (5.60)
– 81 –
Chapter 5. Polynomial and analytic dependence on spectral parameter
Similarly as in Example 5.2.2, see (5.20), we define the 2n × 2n matrices
Tk(λ) :=
(
0 Ak
−I Ck − λW[1]
k
Ak
)
and Tk(λ) :=
(
0 Ak
−I Ck − λW[1]
k
Ak
)
. (5.61)
If ˆz(λ), ˜z(λ) ∈ C([0, ∞)Z)2n solve systems (Qλ) and (Qλ), respectively, then by using the
inverse formula in (5.58) we get that the multiplication of ˆzk+1(λ) by T
∗
k (¯λ)J yields the
backward shift in the second component of ˆzk+1(λ) and the multiplication of ˜zk+1(λ) by
T
∗
k (¯λ)J yields the backward shift in the second component of ˜zk+1(λ), i.e.,
ˆz[s]
k
(λ) := (ˆx∗
k+1(λ), ˆu∗
k(λ))∗
= T
∗
k (¯λ)J ˆzk+1(λ),
˜z[s]
k
(λ) := (˜x∗
k+1(λ), ˜u∗
k(λ))∗
= T
∗
k (¯λ)J ˜zk+1(λ),



(5.62)
compare with (5.23). The same notation will be also used for matrix-valued solutions, in
particular for the fundamental matrices of systems (Qλ) and (Qλ). By similar calculations
as in (5.24) we obtain for any solution Z(λ) ∈ C([0, ∞)Z)2n×m of system (Qλ) and any
solution Z(λ) ∈ C([0, ∞)Z)2n×m of system (Qλ) the Lagrange-type identity
[
Z
∗
k(λ)JZk(ν)
]
= (¯λ − ν)Z
[s]∗
k
(λ)Wk Z
[s]
k
(ν), (5.63)
where we employed the identities in (5.57) and the notation from (5.62). In addition, by
the summation of both sides of (5.63) we get
Z
∗
k+1(λ)JZk+1(ν) = Z
∗
0(λ)JZ0(ν) + (¯λ − ν)
k∑
j=0
Z
[s]∗
j
(λ)Wj Z
[s]
j
(ν), (5.64)
compare with (5.10). In the following result we use the space ℓ2
W
defined in (5.41).
Theorem 5.4.5. Let (5.47) hold. If there exists λ0 ∈ C such that all solutions of systems (Qλ0
)
and (Qλ0
) belong to ℓ2
W
, then all solutions of systems (Qλ) and (Qλ) belong to ℓ2
W
for any λ ∈ C.
Proof. Let the assumptions be satisfied for λ0 ∈ C and λ ∈ CK{λ0} be fixed. For ν ∈ {λ, λ0}
we denote by (ν) ∈ C([0, ∞)Z)2n×2n and (ν) ∈ C([0, ∞)Z)2n×2n the fundamental matrices
of systems (Qν) and (Qν), respectively, such that 0(ν) = I = 0(ν). First we prove that all
solutions of system (Qλ) belong to ℓ2
W
. Since k(λ) and k(λ0) are invertible on [0, ∞)Z,
there exists Ω ∈ C([0, ∞)Z)2n×2n such that
k(λ) = k(λ0)Ωk for all k ∈ [0, ∞)Z, (5.65)
Then by a direct calculation we obtain that Ωk satisfies the recurrence relation
Ωk+1 =
[
I + (λ − λ0)ϒk
]
Ωk, where ϒk := 
−1
k+1(λ0)
[
S
[1]
k + (λ + λ0)S
[2]
k
]
k(λ0). (5.66)
Since 
−1
k+1(λ0) = 
−1
k (λ0)S
−1
k (λ0), we have
det
[
I + (λ − λ0)ϒk
]
= det
[

−1
k (λ0)S
−1
k (λ0)Sk(λ)k(λ0)
] (5.59)
= det
(
S
[0]
k
)−1
× det S
[0]
k = 1,
i.e., the matrix I + (λ − λ0)ϒk is invertible on [0, ∞)Z. By using (5.50) and the n × n block
structure of the matrices Sk(λ0), S
[1]
k , S
[2]
k , Tk(λ0), Tk(λ0), Wk, we get the identity
[
S
[1]
k + (λ0 + ¯λ0)S
[2]
k
]
JS
∗
k(¯λ0) = −Tk(¯λ0)Wk T
∗
k (¯λ0). (5.67)
– 82 –
5.4. Special quadratic dependence and limit circle case
Moreover, a simple calculation yields
S
[1]
k + (λ + λ0)S
[2]
k = Tk
[
S
[1]
k + (λ0 + ¯λ0)S
[2]
k
]
, where Tk :=
(
I 0
(¯λ0 − λ)W[1]
k
I
)
. (5.68)
Hence by using (5.67) and (5.68) in the definition of ϒk in (5.66) and then applying (5.62)
we can equivalently expressed ϒk as
ϒk = Q−1
k 
∗
k+1(λ0)J∗
Tk(¯λ0)Wk T
∗
k (¯λ0)Jk+1(λ0)
(5.62)
= Q−1
k 
[s]∗
k (λ0)Wk 
[s]
k (λ0), (5.69)
where we put
Qk := −
∗
k+1(λ0)JT−1
k k+1(λ0). (5.70)
Since det Tk ≡ 1, we get for any k ∈ [0, ∞)Z the equality
det Qk = det 
∗
k+1(λ0) × det k+1(λ0) = det 
∗
k(λ0) × det k(λ0) × det S
∗
k(λ0) × det Sk(λ0)
(5.60)
= det 
∗
k(λ0) × det k(λ0) = · · · = det 
∗
0(λ0) × det 0(λ0) = 1.
Now we show that there exists κ > 0 such that
||Q−1
k ||σ ≤ κ < ∞ on [0, ∞)Z. (5.71)
Since Wk ≥ 0, the Cauchy–Schwarz and arithmetic-geometric mean inequalities yield
|ξ∗
Wk ζ| ≤ (ξ∗
Wk ξ)1/2
(ζ∗
Wk ζ)1/2
≤ 1
2 (ξ∗
Wk ξ + ζ∗
Wk ζ)
for any ξ, ζ ∈ C2n and k ∈ [0, ∞)Z. Hence, for any sequences ˆz, ˜z ∈ ℓ2
W
we have
∞∑
k=0
˜z[s]∗
k
Wk ˆz[s]
k
≤
∞∑
k=0
˜z[s]∗
k
Wk ˆz[s]
k
≤ 1
2
∞∑
k=0
(˜z[s]∗
k
Wk ˜z[s]
k
+ ˆz[s]∗
Wk ˆz[s]
k
) < ∞.
The last inequality with ˆz and ˜z being the columns of (λ0) and (λ0), respectively, and
the assumption that all solutions of systems (Qλ0
) and (Qλ0
) belong to ℓ2
W
imply that there
exists ε > 0 such that
∞∑
k=0

[s]∗
k (λ0)Wk 
[s]
k (λ0) σ
(1.7)
≤
∞∑
k=0

[s]∗
k (λ0)Wk 
[s]
k (λ0) 1
≤ ε < ∞. (5.72)
Since JT−1
k
= J − (¯λ0 − λ) diag
{
W[1]
k
, 0
}
, the sequence Qk in (5.70) can be written as
Qk = 
∗
k+1(λ0)
[
(¯λ0 − λ) diag
{
W[1]
k
, 0
}
− J
]
k+1(λ0)
(5.64)
= −J + 2i im(λ0)
k∑
j=0

[s]∗
j (λ0)Wj 
[s]
j (λ0)
+ (¯λ0 − λ)
[s]∗
k (λ0) diag
{
W[1]
k
, 0
}

[s]
k (λ0). (5.73)
From the unitary invariance of the spectral norm, assumption (5.47), and the estimate
in (5.72) we conclude that there exists τ > 0 such that

[s]∗
k (λ0) diag
{
W[1]
k
, 0
}

[s]
k (λ0) σ
≤ 
[s]∗
k (λ0)Wk 
[s]
k (λ0) σ
≤ τ < ∞ (5.74)
– 83 –
Chapter 5. Polynomial and analytic dependence on spectral parameter
for all k ∈ [0, ∞)Z. Upon taking the matrix norm in (5.73) we obtain for any k ∈ [0, ∞)Z that
||Qk ||σ ≤ ||J||σ + 2| im(λ0)|
k∑
j=0

[s]∗
j (λ0)Wj 
[s]
j (λ0) σ
+ |¯λ0 − λ| 
[s]∗
k (λ0) diag
{
W[1]
k
, 0
}

[s]
k (λ0) σ
(5.72),(5.74)
≤ ω < ∞,
where ω := ||J||σ + 2| im(λ0)|ε + |¯λ0 − λ|τ. Therefore the matrix Qk is bounded on [0, ∞)Z
with det Qk ≡ 1. This implies that Q−1
k
is also bounded on [0, ∞)Z, i.e., inequality (5.71)
holds. By combining the submultiplicative property of the spectral norm and (5.69),
(5.71), (5.72) we then obtain
∞∑
k=0
||ϒk ||σ ≤
∞∑
k=0
||Q−1
k ||σ × 
[s]∗
k (λ0)Wk 
[s]
k (λ0) σ
≤ κε < ∞.
Hence the same calculation as in (4.21), see also Proposition 1.1.4, implies that the fundamental
matrix Ωk of system (5.66) satisfies
||Ωk ||σ ≤ ρ for all k ∈ [0, ∞)Z and some ρ > 0. (5.75)
With Kk(λ) := 
[s]∗
k (λ)Wk 
[s]
k (λ) we obtain from (5.65) and (5.47) that
Kk(λ) = Ω
∗
k+1 
[s]∗
k (λ0) diag
{
W[1]
k
, 0
}

[s]
k (λ0)Ωk+1 + Ω
∗
k 
[s]∗
k (λ0) diag
{
0, W[2]
k
}

[s]
k (λ0)Ωk
≤ Ω
∗
k+1 Kk(λ0)Ωk+1 + Ω
∗
k Kk(λ0)Ωk. (5.76)
This implies through (5.75) and the submultiplicativity, self-adjointness, and unitary
invariance of the spectral norm that
∞∑
k=0
||Kk(λ)||σ
(5.76)
≤
∞∑
k=0
(
Ω
∗
k+1 Kk(λ0)Ωk+1 σ
+ Ω
∗
k Kk(λ0)Ωk σ
)
≤
∞∑
k=0
(
Ωk+1
2
σ
+ Ωk
2
σ
)
× ||Kk(λ0)||σ
(5.75)
≤ 2ρ2
∞∑
k=0
||Kk(λ0)||σ < ∞,
because all solutions of system (Qλ0
) belong to ℓ2
W
. This shows that all solutions of
system (Qλ) belong to ℓ2
W
as well. Analogously, by switching the roles of systems (Qλ)
and (Qλ), we prove that all solutions of system (Qλ) belong to ℓ2
W
. Since λ ∈ CK{λ0} was
chosen arbitrarily, the proof is complete. ■
Proof of Theorem 5.4.1. The statement of Theorem 5.4.1 follows immediately from Theorem
5.4.5, when it is applied in the case S
[0]
k ≡ S
[0]
k := S[0]
k
, i.e., when all systems (Qλ), (Qλ),
and (Qλ) are equal. ■
Now we give sufficient conditions in terms of the coefficients, which guarantee that all
solutions of systems (Qλ) and (Qλ) belong to the space ℓ2
W
. Similarly as in Corollary 4.2.3,
the matrix norm ||·||1 utilized in (5.77) below can be replaced by any other matrix norm
because of their equivalence.
– 84 –
5.4. Special quadratic dependence and limit circle case
Corollary 5.4.6. Let (5.47) hold and assume that
∞∑
k=0
S
[0]
k − I 1
< ∞,
∞∑
k=0
S
[0]
k − I 1
< ∞, and
∞∑
k=0
||Wk ||1 < ∞. (5.77)
Then all solutions of systems (Qλ) and (Qλ) belong to ℓ2
W
for any λ ∈ C.
Proof. By Theorem 5.4.5 it suffices to show that there exists λ0 ∈ C such that
∞∑
k=0

[s]∗
k (λ0)Wk 
[s]
k (λ0) 1
< ∞ and
∞∑
k=0

[s]∗
k (λ0)Wk 
[s]
k (λ0) 1
< ∞. (5.78)
We show that these inequalities are satisfied for λ0 = 0. Since Sk(0) = S
[0]
k , we obtain
from the first condition in (5.77) and Proposition 1.1.4 that there exist ε > 0 such that
k(0) 1
≤ ε < ∞ for all k ∈ [0, ∞)Z, where (0) ∈ C([0, ∞)Z)2n×2n represents a fundamental
matrix of system (Q0). Moreover, the second condition in (5.77) implies that there exists
ρ > 0 such that Ak − I 1
+ Ck 1
≤ ρ < ∞ for all k ∈ [0, ∞)Z. Hence from (5.61) we get
T
∗
k (0)J 1
=
(
0 0
−C
∗
k A
∗
k − I
)
+ I2n
1
≤
(
0 0
−C
∗
k A
∗
k − I
)
1
+ ||I2n ||1
≤ Ak − I 1
+ Ck 1
+ 2n ≤ κ < ∞ for all k ∈ [0, ∞)Z,
where we used the self-adjointness of the norm ||·||1 and put κ := 2n + ρ. Since we have

[s]
k (0) = T
∗
k (0)Jk+1(0) by (5.62), the last condition in (5.77) yields the estimate
∞∑
k=0

[s]∗
k (0)Wk 
[s]
k (0) 1
≤
∞∑
k=0

[s]
k (0)
2
1
× ||Wk ||1 ≤
∞∑
k=0
T
∗
k (0)J
2
1
× k+1(0)
2
1
× ||Wk ||1
≤ κ2
ε2
∞∑
k=0
||Wk ||1 < ∞.
In a similar way we prove also the second inequality in (5.78). Therefore all solutions of
systems (Q0) and (Q0) belong to ℓ2
W
, and so the statement follows from Theorem 5.4.5. ■
If the matrices Ak and Ak are invertible on [0, ∞)Z, then systems (Qλ) and (Qλ) are
equivalent with the pair of the first order difference systems
ˆzk(λ) =
[
Hk + λJWk
]
ˆz[s]
k
(λ) and ˜zk(λ) =
[
Hk + λJWk
]
˜z[s]
k
(λ),
where W is from (5.47), the coefficient matrix H ∈ C([0, ∞)Z)2n×2n has the form
Hk :=


I − A
−1
k A
−1
k Bk
Ck A
−1
k Dk − Ck A
−1
k Bk − I


(5.50)
=


I − A
−1
k A
−1
k Bk
Ck A
−1
k A
∗−1
k − I

 ,
and H ∈ C([0, ∞)Z)2n×2n is given analogously, cf. Examples 5.2.2 and 5.2.3. Especially, if
we take Ak ≡ Ak, then the first identities in (5.50) and (5.55) imply that Hk = JH∗
k
J for
all k ∈ [0, ∞)Z. In this case we obtain from Theorem 5.4.5 the following generalization
of [142, Theorem 5.5] for two non-Hermitian linear Hamiltonian difference systems
ˆzk(λ) =
[
Hk + λWk
]
ˆz[s]
k
(λ), (Hλ)
˜zk(λ) =
[
JH
∗
k J + λWk
]
˜z[s]
k
(λ), (Hλ)
– 85 –
Chapter 5. Polynomial and analytic dependence on spectral parameter
where we put
Hk :=
(
Ak Bk
Ck −A
∗
k
)
and Wk :=
(
0 W
[2]
k
−W
[1]
k 0
)
(5.79)
with A, B, C, W
[1]
, W
[2]
∈ C([0, ∞)Z)n×n being such that I − Ak is invertible, W
[1]
k , W
[2]
k are
Hermitian, and −JWk ≥ 0 on [0, ∞)Z. Note that we have (Hλ)=(Hλ)=(2.6) if Bk = B
∗
k = Bk
and Ck = C
∗
k = Ck.
Corollary 5.4.7. Let H, W ∈ C([0, ∞)Z)2n×2n be as in (5.79). If there exists λ0 ∈ C such that
all solutions of systems (Hλ0
) and (Hλ0
) belong to ℓ2
W
with W := −JWk, then all solutions of
systems (Hλ) and (Hλ) belong to ℓ2
W
for any λ ∈ C.
Finally, we give an example illustrating the result of Theorem 5.4.5.
Example 5.4.8. Let κ ≥ 1 be fixed and q, r ∈ C([0, ∞)Z)1 be nonnegative sequences such
that F(κ) :=
∑∞
k=0 Fk(κ) < ∞ and G(κ) :=
∑∞
k=0 Gk(κ) < ∞, where
Fk(κ) := κ8k+2
q4k + κ8k+4
r4k+2 + κ8k+6
r4k+3 + κ8k+8
q4k+3,
Gk(κ) := κ8k
r4k + κ8k+2
r4k+1 + κ8k+4
q4k+1 + κ8k+6
q4k+2.



(5.80)
For example, we may choose
qk = rk := 1/(ck
κ3k
) with c > 1/κ,
because in this case the numbers F(κ) and G(κ) are multiples of the convergent series∑∞
k=0 1/(cκ)4k. By substituting 1/κ instead of κ in (5.80) we can see that the series F(1/κ)
and G(1/κ) are convergent as well.
Let us consider systems (Qλ) and (Qλ) with n = 1 and with the following coefficients:
(i) for all k ∈ [0, ∞)Z we put W[1]
k
:= qk and W[2]
k
:= rk, (ii) for k ∈ [0, ∞)Z even (k = 2j)
we set Ak = Dk := (−1)j/κ, Bk = Ck := 0 and Ak = Dk := (−1)jκ, Bk = Ck := 0, while for
k ∈ [0, ∞)Z odd (k = 2j + 1) we define Ak = Dk := 0, Bk = −Ck := (−1)j/κ and Ak = Dk := 0,
Bk = −Ck := (−1)jκ. This means that
S
[0]
k = (1/κ)Jk
, S
[0]
k = κJk
, and Wk = diag
{
W[1]
k
, W[2]
k
}
= diag{qk, rk} ≥ 0. (5.81)
Then conditions (5.47) and (5.49) are satisfied. We show that all solutions of systems (Q0)
and (Q0) with the coefficients specified above belong to the corresponding space ℓ2
W
. The
fundamental matrices k(0) and k(0) of systems (Q0) and (Q0) determined by the initial
conditions 0(0) = I = 0(0) are equal to
k(0) = (1/κk
)J(k2−k)/2
and k(0) = κk
J(k2−k)/2
for all k ∈ [0, ∞)Z.
From (5.61) we obtain Tk(0) =
(
0 Ak
−I Ck
)
and Tk(0) =
(
0 Ak
−I Ck
)
, and thus for k ∈ [0, ∞)Z we get

[s]
k (0) = T
∗
k (0)Jk+1(0) =
((
ˆz[1]
k
)[s]
,
(
ˆz[2]
k
)[s]
)
=



(−1)j
κ4j+1


1 0
0 κ

 , k = 4j,
(−1)j
κ4j+2


0 1
0 κ

 , k = 4j + 1,
(−1)j+1
κ4j+3


0 1
κ 0

 , k = 4j + 2,
(−1)j+1
κ4j+4


1 0
−κ 0

 , k = 4j + 3,
– 86 –
5.5. Bibliographical notes
where
(
ˆz[1]
k
)[s]
and
(
ˆz[2]
k
)[s]
mean the partial shift applied to ˆz[1]
k
and ˆz[2]
k
, respectively, i.e.,
(
ˆz[1]
k
)[s]
=
(
ˆx[1]
k+1
, ˆu[1]
k
)⊤
and
(
ˆz[2]
k
)[s]
=
(
ˆx[2]
k+1
, ˆu[2]
k
)⊤
. Similarly, the matrix

[s]
k (0) = T
∗
k (0)Jk+1(0) =
((
˜z[1]
k
)[s]
,
(
˜z[2]
k
)[s]
)
has the same form as 
[s]
k (0) above but with κ replaced by 1/κ. By direct calculations we
then have
∞∑
k=0
(
ˆz[1]
k
)[s]∗
Wk
(
ˆz[1]
k
)[s]
= F(1/κ) < ∞,
∞∑
k=0
(
ˆz[2]
k
)[s]∗
Wk
(
ˆz[2]
k
)[s]
= G(1/κ) < ∞,
∞∑
k=0
(
˜z[1]
k
)[s]∗
Wk
(
˜z[1]
k
)[s]
= F(κ) < ∞,
∞∑
k=0
(
˜z[2]
k
)[s]∗
Wk
(
˜z[2]
k
)[s]
= G(κ) < ∞.
This shows that ˆz[1]
, ˆz[2]
, ˜z[1]
, ˜z[2]
∈ ℓ2
W
, i.e., the assumptions of Theorem 5.4.5 are satisfied
with λ0 = 0. Therefore, by this theorem, all solutions of systems (Qλ) and (Qλ) with (5.81)
belong to ℓ2
W
for any λ ∈ C. Observe that the statement of Corollary 5.4.6 cannot be
applied, because
∑∞
k=0 S
[0]
k −I 1
= ∞ =
∑∞
k=0 S
[0]
k −I 1
, i.e., the first two conditions in (5.77)
are now violated. Note also that systems (Qλ) and (Qλ) in this example cannot be written
as systems (Hλ) and (Hλ), respectively, because the coefficient matrices Ak = 0 = Ak for
k ∈ [0, ∞)Z odd are singular. ▲
Remark 5.4.9. We note that Example 5.4.8 with κ = 1 illustrates the application of Theorem
5.4.1, since in this case the systems (Qλ) and (Qλ) coincide.
5.5 Bibliographical notes
The results of this chapter were published in [A17] and their generalization to symplectic
systems on time scales was established in [A20]. More precisely, Theorem 5.1.3, Example
5.2.3, and Section 5.4 were published only for systems on time scales and they are
explicitly presented for the first time in the discrete case, while Examples 5.3.9 and 5.4.8
are taken almost verbatim from [A20]. However it is worth noticing that the results
of [A17,A20] were stated without the shift in the definition of the space ℓ2
Ψ(λ)
. A generalization
of the invariance of the limit circle case for system (Sλ) with the special polynomial
dependence on λ described in Example 5.2.3 will be a subject of our future research.
– 87 –
Chapter 5. Polynomial and analytic dependence on spectral parameter
– 88 –
Chapter 6
Nohomogeneous problem and
maximal and minimal linear
relations
To compare the discrete with the continuous, to search for analogies
between them, and ultimately to effect their unification, are patterns of
mathematical development that did not begin with Zeno, and certainly
did not end with Leibnitz and Newton, nor even with Riemann and
Stieltjes. Such a pattern of investigation is especially appropriate to the
theory of boundary problems, for which the discrete and the continuous
pervade both physical origins and mathematical methods.
Frederick Valentine Atkinson, see [9, pg. v]
In the last two chapters we return to the case of linear dependence on the spectral
parameter and focus on the discrete symplectic systems from the “operator-theoretic”
point of view. To the best of the author’s knowledge, this direction in the theory of discrete
symplectic systems was completely untouched before the publications [A18,A21], see also
the bibliographical notes in Sections 6.5 and 7.4. However, instead of system (Sλ) with
the matrices 5k, Sk, Vk, k we now consider the underlying discrete symplectic system in
the so-called time-reversed form with the matrices pk, Sk, Vk, ψk (to avoid any confusion
we use two different fonts), i.e.,
zk(λ) = pk(λ)zk+1(λ) with pk(λ) := Sk + λVk, (Sλ)
where λ ∈ C is the same as in the previous chapters and Sk, Vk ∈ C2n×2n are such that
S∗
k JSk = J, S∗
k JVk is Hermitian, V∗
k JVk = 0, and ψk := JSk JV∗
kJ ≥ 0, (6.1)
where the matrix J is (again) as in (1.12). This change is mainly motivated by the absence
of the shift in the definition of the associated semi-inner product and semi-norm, see (6.8)
and compare with (2.54) and (2.26). Consequently, it produces a more natural form of
the Green function associated with nonhomogeneous discrete symplectic systems, see
Lemma 6.3.2, and allows us to associate with system (Sλ) a densely defined operator, see
Theorem 6.4.5 and compare with [132].
Regardless the transition from system (Sλ) to (Sλ), the results of the previous chapters
remain valid also for system (Sλ) with the changes given for the definition of the semiinner
product and semi-norm. More precisely, one easily observes that the first three
identities in (6.1) are the same as in (2.1). Therefore (again) the matrix Sk is symplectic,
– 89 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
for any λ ∈ C we have
p∗
k(¯λ)Jpk(λ) = J, p−1
k (λ) = −Jp∗
k(¯λ)J, | det pk(λ)| = | det Sk | = 1, (6.2)
and the identities for the matrices Sk, Vk, pk(λ) can be modified similarly as in Lemma 2.1.1,
see Lemma 6.1.1. Consequently system (Sλ) is equivalent to system (Sλ), where we put
5k(λ) := p−1
k
(λ), i.e., Sk := −JS∗
k
J and Vk := −JV∗
k
J. But in that case we obtain
z∗
k(λ)ψk zk(λ) = z∗
k+1(λ)p∗
k(λ)ψk pk(λ)zk+1(λ) = −z∗
k+1(λ)V∗
k JSk zk+1(λ)
= z∗
k+1(λ)JVk JS∗
k Jzk+1(λ) = z∗
k+1(λ) k zk+1(λ), (6.3)
which among others justifies the replacement of k by ψk. On the other hand, systems (Sλ)
and (Sλ) lead to different spaces of square summable sequences, see (2.53) and (6.9).
Finally, we remark that although the presence of the shift in the semi-inner product could
be suppressed similarly as in [A17], the approach based on the time-reversed system
seems to be more natural in the present situation and it is, in addition, traditionally used in
connection with the second order Sturm–Liouville difference equations, see e.g. [96,147].
Moreover, while for the analyses of the square summable solutions developed in the
previous chapters we it was justifiable to “ignore” finite discrete intervals, in this and the
following chapters we concern with system (Sλ) on a discrete interval IZ, which is finite
or unbounded above, i.e., IZ = [0, N + 1)Z with N ∈ N ∪ {0, ∞}.
Given the inherent positive semidefiniteness of the sequence ψ defined in (6.1), it is
reasonable to consider the construction of operators in connection with system (Sλ), their
extensions and their spectral theory. However we will see that the natural map associated
with system (Sλ) does give rise only to a multivalued or non-densely defined operator,
see Section 6.4. Hence the approach dealing with linear relations instead of operators is
utilized as it provides powerful tools for the investigation of multivalued linear operators
in a Hilbert space, especially for non-densely defined linear operators. The beginning of
the general theory of linear relations can be traced back to [8], where it was established as
a generalization of the results carried out in [167], see also [42,45,46,82,143,144,146,175]
and the references therein. Moreover, a short introduction to this theory is provided
in the Appendix of this thesis and the reader should be acquainted with its content
before reading Section 6.4. The study of linear relations in connection with the linear
Hamiltonian differential system from (2.5) was initiated in [128] and further developed,
e.g., in [13,83,104,116]. One of the first occurrences of this concept in the discretetheory can
be found in [28,29] for the second order Sturm–Liouville difference equations, while their
extension to the linear Hamiltonian difference systems was given recently in [79,134,135,
155,161], compare with [142,163]. Hence, in this chapter we aim to associate the minimal
and maximal linear relations to the (time-reversed) discrete symplectic systems and to
establish their fundamental properties in analogy with [116, Section 2] for system (2.5)
and with [134, Section 5] for system (2.6).
The rest of this chapter is organized as follows. In the following section we give some
basic properties of system (Sλ). In Section 6.2 we focus on the definiteness (or the strong
Atkinson) condition for system (Sλ), which plays a crucial role in the present theory,
and derive some equivalent characterizations. A nonhomogeneous discrete symplectic
system is investigated in Section 6.3. Concluding with Section 6.4, the maximal and
minimal linear relations associated with the (time-reversed) discrete symplectic systems
are introduced and their fundamental properties, such as a relationship between the
deficiency indices of the minimal relation in a suitable Hilbert space and the number of
– 90 –
6.1. Preliminaries
square summable solutions of system (Sλ), are established. In this final section we also
present a sufficient condition guaranteeing the existence of a densely defined operator
associated with the time-reversed discrete symplectic system.
6.1 Preliminaries
The conditions for the coefficients Sk and Vk of system (Sλ) given in (6.1) can be expressed
in several equivalent forms, which are summarized in the following statement, compare
with Lemma 2.1.1. Moreover, the same arguments as in (2.1.3) can be used for the
calculation of | det pk(λ)|.
Lemma 6.1.1. Let n ∈ N be given. For any k ∈ [0, ∞)Z the following conditions are equivalent.
(i) The matrices Sk and Vk satisfy the first three conditions in (6.1), i.e., S∗
k
JSk = J, S∗
k
JVk is
Hermitian, and V∗
k
JVk = 0.
(ii) The matrix pk(λ) defined in (Sλ) satisfies the first equality in (6.2), i.e., p∗
k
(¯λ)Jpk(λ) = J,
for all λ ∈ C.
(iii) The matrices Sk and Vk satisfy Sk JS∗
k
= J and Vk JV∗
k
= 0, and Vk JS∗
k
is Hermitian.
(iv) The matrix pk(λ) in (Sλ) satisfies pk(λ)Jp∗
k
(¯λ) = J for all λ ∈ C.
Moreover, if any of conditions (i), (ii), (iii) or (iv) holds, then
pk(λ) = (I − λJψk)Sk, ψ∗
k = ψk, ψk Jψk = 0, and (I − λJψk)−1
= (I + λJψk), (6.4)
where ψk is defined (without the requirement of the positive semidefiniteness) in (6.1). Consequently,
| det pk(λ)| = | det Sk(λ)| = 1 for all λ ∈ C.
The invertibility of all matrices pk(λ) guarantees the (global) existence and uniqueness
of a solution of any initial value problem associated with system (Sλ). Especially, if
z(λ) ∈ C(I+
Z )2n solves system (Sλ) and satisfies zs(λ) = 0 for some s ∈ I+
Z , then z(λ) is only
a trivial solution, i.e., zk(λ) = 0 for all k ∈ I+
Z .
The latter lemma also establishes the correspondence between the matrix pairs {S, V}
and {S, ψ}. More precisely, if Sk, Vk ∈ C2n×2n satisfies the first three conditions in (6.1),
then the equalities in (6.4) hold. On the other hand, if Sk, ψk ∈ C2n×2n are such that Sk
satisfies the first relation in (6.1), ψ∗
k
= ψk, and ψk Jψk = 0, then the second and third
conditions in (6.1) hold with Vk := J∗ψk Sk = −JψkSk. Simultaneously, from (6.4) one can
easily conclude that system (Sλ) can be equivalently written as
J
(
zk(λ) − Sk zk+1(λ)
)
= λψk zk(λ),
which gives rise to a natural linear map L : C(I+
Z )2n×m → C(IZ)2n×m associated with
system (Sλ), namely
L (z)k := J(zk − Sk zk+1), (6.5)
where m ∈ N corresponds to the dimension of z and typically we consider m = 1. Hence
system (Sλ) is equivalent to
L (z(λ))k = λψk zk(λ), k ∈ IZ. (6.6)
From this point of view, the second approach seems to be more natural in the present
situation, i.e., to “fix” the sequences of matrices S, ψ ∈ C(IZ)2n×2n such that
S∗
k JSk = J, ψ∗
k = ψk, ψk Jψk = 0, and ψk ≥ 0 for all k ∈ IZ. (6.7)
In any case, throughout Chapters 6 and 7 we employ the following notation.
– 91 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
Notation 6.1.2. The number n ∈ N is fixed, the discrete interval IZ is given either finite or
infinite, i.e., IZ = [0, N + 1)Z with N ∈ N ∪ {0, ∞}, and the sequences S, V, ψ ∈ C(IZ)2n×2n
are such that the conditions in (6.1) are satisfied for all k ∈ IZ.
With respect to the fact that ψ ∈ C(IZ)2n×2n is a sequence with positive semidefinite
terms, we define a semi-inner product on C(I+
Z )2n by
⟨z, v⟩ψ :=
∑
k∈I
z∗
k ψk vk, (6.8)
where z, v ∈ C(I+
Z )2n. Here we are intentionally sloppy when suppressing the subscript Z
in the notation of the discrete interval IZ in the sum on the right-hand side of the latter
identity. The same convention is applied throughout the last two chapters. The associated
linear space of all square summable sequences is denoted by
l2
ψ(IZ) = l2
ψ :=
{
z ∈ C(I+
Z )2n
| ||z||ψ < ∞
}
, (6.9)
where ||·||ψ :=
√
⟨·, ·⟩ψ denotes the associated semi-norm.
The equivalent expression of system (Sλ) given in (6.6) also justifies the following form
of the associated nonhomogeneous system
zk(λ) = pk(λ)zk+1(λ)−Jψk fk or equivalently L (z(λ))k = λψk zk(λ)+ψk fk, k ∈ IZ, (S
f
λ
)
where z(λ), f ∈ C(I+
Z )2n×m with m ∈ N being the same as discussed when defining the
linear map L in (6.5). For simplicity, (S
g
ν ) refers to the nonhomogeneous system of the
form given in (S
f
λ
) with λ replaced by ν and f replaced by g. In addition, instead of (S0
λ
)
we write only (Sλ) and for this system we apply the same convention, i.e., by (Sν) we
denote the system of the form given in (Sλ) with the parameter λ replaced by ν. Let us
point out that the equivalent expression given in (S
f
λ
) will play a key role in Section 6.4,
where we sometimes write only L (z(λ)) = λψz(λ)+ψf and L (z) = ψf with the meaning
L (z(λ))k = λψk zk(λ)+ψk fk and L (z)k = ψk fk for all k ∈ IZ, respectively. For convenience,
we also put L ∗(z)k := [L (z)k]∗.
The next result presents an absolutely essential tool used throughout this chapter,
compare with Lemma 2.1.5 and Theorem 2.1.7.
Theorem 6.1.3 (Extended Lagrange identity). Let λ, ν ∈ C, m ∈ N, and f, g ∈ C(IZ)2n×m be
given. If the sequences Z(λ), Z(ν) ∈ C(I+
Z )2n×m solve systems (S
f
λ
) and (S
g
ν ) on IZ, respectively,
then for any s, t ∈ IZ with s ≤ t we have
[
Z∗
k(λ)JZk(ν)
]
= (¯λ − ν)Z∗
k(λ)ψk Zk(ν) + f∗
k ψk Zk(ν) − Z∗
k(λ)ψk gk, (6.10)
Z∗
k(λ)JZk(ν)
t+1
s
=
t∑
k=s
{
(¯λ − ν)Z∗
j (λ)ψj Zj(ν) + f∗
j ψj Zj(ν) − Z∗
j (λ)ψj gj
}
. (6.11)
Especially, if ν = ¯λ and fk ≡ gk ≡ 0 on IZ, we have the Wronskian-type identity
Z∗
k(λ)JZk(¯λ) ≡ Z∗
0(λ)JZ0(¯λ) on I+
Z . (6.12)
Proof. Let Z(λ), Z(ν) ∈ C(I+
Z )2n×m solve systems (S
f
λ
) and (S
g
ν ). Since by (6.4) we have
p∗−1
k
(λ)Jp−1
k
(ν) = J + (¯λ − ν)ψk, we see that
[
Z∗
k(λ)JZk(ν)
]
=
[
Zk(λ) + Jψk fk
]∗
p∗−1
k (λ)Jp−1
k (ν)
[
Zk(ν) + Jψkgk
]
− Z∗
k(λ)JZk(ν)
= Z∗
k(λ)
[
p∗−1
k (λ)Jp−1
k (ν) − J
]
Zk(ν) + f∗
k ψk Zk(ν) − Z∗
k(λ)ψk gk
= (¯λ − ν)Z∗
k(λ)ψk Zk(ν) + f∗
k ψk Zk(ν) − Z∗
k(λ)ψk gk.
– 92 –
6.2. Definiteness condition
Identities (6.11) and (6.12) are only simple consequences of the latter equality. ■
Let us note that, by using the equivalent expression of system (S
f
λ
), identity (6.11) can
be also written as
(
Z(λ), Z(ν)
)
k
t+1
s
=
t∑
k=s
{
L ∗
(Z(λ))k Zk(ν) − Z∗
k(λ)L (Z(ν))k
}
, (6.13)
where we use for any Z, Z ∈ C(I+
Z )2n×m and k ∈ I+
Z the notation
(Z, Z)k := Z∗
k JZk.
Moreover, under the assumptions of Theorem 6.1.3 with m = 1, Z(λ) = z, Z(ν) = v,
λ = 0 = ν, s = 0, and t = N we get from (6.11) and (6.8) that
(z, v)k
N+1
0
= ⟨ f, v⟩ψ − ⟨z, g⟩ψ, (6.14)
where the left-hand side of (6.14) means limk→∞(z, v)k −(z, v)0 if IZ = [0, ∞)Z. Nevertheless
identity (6.14) and the Cauchy–Schwarz inequality imply that the latter limit exists finite
whenever z, v, f, g ∈ l2
ψ
.
Throughout this and the following chapters we denote by Θ(λ) ∈ C(I+
Z )2n×2n a fundamental
matrix of system (Sλ). If, similarly as in Chapter 2, it is such that
Θ∗
s(¯λ)JΘs(λ) = J, (6.15)
for some s ∈ I+
Z , then as an immediate consequence of (6.12) we have for any k ∈ I+
Z that
Θ∗
k(¯λ)JΘk(λ) = J, Θ−1
k (λ) = −JΘ∗
k(¯λ)J, and Θk(λ)JΘ∗
k(¯λ) = J. (6.16)
Remark 6.1.4. Similarly as for linear Hamiltonian differential systems, there exists a unitary
map Q : C(I+
Z )2n → C(I+
Z )2n preserving the square summability with respect to ψ
and such that system (S
f
0
) can be written in the canonical form, i.e., with S ≡ I. Indeed,
let Θ = Θ(0) denote the fundamental matrix of system (S0) satisfying Θ0 = I. Then it
is invertible for all k ∈ I+
Z with Θ−1
k
= −JΘ∗
k
J and this inverse provides the canonical
transformation, i.e., Qk = Θ−1
k
with Q(z)k := Θ−1
k
zk. Hence system (S
f
0
) is equivalent with
− J vk = ψk gk, k ∈ IZ, (6.17)
where we put vk := Q(z)k, gk := Q(f)k, and ψk := Θ∗
k
ψk Θk, see also [79]. Moreover, one
can easily verify that v ∈ l2
ψ
if and only if z ∈ l2
ψ
. System (6.17) can be seen as a discrete
counterpart of the canonical linear Hamiltonian differential system, i.e., nonhomogeneous
system associated with (2.5), where H(t) ≡ 0, see e.g. [128] or [116, Subsection 2.2] and the
references therein.
6.2 Definiteness condition
In this section we focus on the definiteness condition for system (Sλ), which is closely
related to the strong Atkinson condition applied to all λ ∈ C, see Hypothesis 2.4.11
and Remark 6.2.7. Although it was shown in Chapter 2 that only the weak form of the
Atkinson condition is sufficient for the development of the Weyl–Titchmarsh theory, we
will need the strong version for (at least) two reasons:
– 93 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
(i) in the section devoted to the nonhomogeneous problem we utilize Theorem 2.4.12
in the setting of system (Sλ), see Section 6.3;
(ii) it guarantees a certain (extremely important) uniqueness property for system (S
f
0
),
see Lemmas 6.2.8 and 6.4.3, Theorem 6.4.2, Corollary 6.4.14, and Chapter 7.
Similar treatment in connection with the linear Hamiltonian differential and difference
systems can be found in [13,116] and [134], respectively.
Definition 6.2.1. System (Sλ) is said to be definite on a discrete interval ID
Z ⊆ IZ provided
the interval ID
Z is nonempty and for each λ ∈ C any nontrivial solution z(λ) ∈ C(I+
Z )2n of
system (Sλ), i.e, L (z(λ))k = λψk zk(λ) for all k ∈ IZ, satisfies
∑
k∈ID
z∗
k(λ)ψk zk(λ) > 0. (6.18)
Remark 6.2.2. Alternatively, the definiteness condition for (Sλ) can be stated in the following
way: system (Sλ) is definite on a (nonempty) discrete interval ID
Z ⊆ IZ if, for each
λ ∈ C, every solution z(λ) ∈ C(I+
Z )2n of system (Sλ) for which
∑
k∈ID
z∗
k(λ)ψk zk(λ) = 0, (6.19)
is trivial on ID
Z , i.e., zk(λ) = 0 for all k ∈ ID
Z , and consequently it is trivial on the whole
interval I+
Z . Furthermore, from the assumption of ψk ≥ 0 on IZ we get immediately that
for any discrete interval IZ ⊆ IZ such that ID
Z ⊆ IZ we have
∑
k∈ID
z∗
k(λ)ψk zk(λ) ≤
∑
k∈I
z∗
k(λ)ψk zk(λ). (6.20)
Therefore one easily concludes that the definiteness of system (Sλ) on ID
Z guarantees the
definiteness of (Sλ) on every discrete “interval superset” IZ, especially on IZ. Hence,
definiteness of system (Sλ) on some finite discrete subinterval ID
Z implies, for every λ ∈ C,
that the semi-norm ||·||ψ of any nontrivial solution of system (Sλ) is nonzero. The converse
of this last statement will be established in Lemma 6.2.6 below.
In the next lemma we show that for the definiteness of system (Sλ) on a discrete
interval ID
Z it is not necessary to verify inequality (6.18) for all nontrivial solutions and
every λ ∈ C, but it suffices to do it only for one particular choice of λ ∈ C.
Lemma 6.2.3. System (Sλ) is definite on a discrete interval ID
Z ⊆ IZ if and only if, for some
λ0 ∈ C, each solution z(λ0) ∈ C(I+
Z )2n of system (Sλ0
) satisfying
∑
k∈ID
z∗
k(λ0)ψk zk(λ0) = 0 (6.21)
is trivial on ID
Z , i.e., zk(λ0) = 0 for all k ∈ ID
Z .
Proof. We begin by assuming that for some λ0 ∈ C each solution z(λ0) of system (Sλ0
)
satisfying (6.21) is necessarily trivial on ID
Z , i.e., zk(λ0) = 0 for all k ∈ ID
Z . Let λ ∈ C be
arbitrary and z(λ) be a solution of system (Sλ) such that (6.19) holds. Given Remark 6.2.2,
it suffices to show that z(λ) is also trivial on ID
Z . Since ψk zk(λ) = 0 for all k ∈ ID
Z , we see
by the equivalent expression given in (6.6) that z(λ) solves also system (Sλ0
) on ID
Z . Then,
by the assumed definiteness of system (Sλ0
) on ID
Z , condition (6.19) indeed implies that
zk(λ) = 0 for all k ∈ ID
Z . The converse is trivial. ■
– 94 –
6.2. Definiteness condition
Example 6.2.4. The discrete symplectic systems (or their time-reversed form) investigated
in Examples 2.5.1–2.5.3 are not definite, because they possess nontrivial solutions with
the semi-norm equal to zero. On the other hand, we can provide a simple example of
system (Sλ) being definite on some nontrivial discrete interval ID
Z ⊆ IZ; cf. Theorem 6.2.5.
Let us consider the following system
(
xk(λ)
uk(λ)
)
=
(
1 −1/pk+1
−qk + λwk 1 + (qk − λwk)/pk+1
) (
xk+1(λ)
uk+1(λ)
)
, k ∈ IZ, (6.22)
where p ∈ C(I+
Z K{0}) and q, w ∈ C(IZ) are (only) real-valued sequences such that wk ≥ 0
and pk+1 0 for all k ∈ IZ. Then the conditions in (6.1) are satisfied with
Sk =
(
1 −1/pk+1
−qk 1 + qk/pk+1
)
, Vk =
(
0 0
wk −wk/pk+1
)
, and ψk =
(
wk 0
0 0
)
,
see also (6.25). If wk > 0 for at least two consecutive points of IZ, i.e., for k ∈ [a, b]Z with
a, b ∈ IZ and a < b, then system (6.22) is definite on [a, b]Z, and thus, by Remark 6.2.2,
also on any discrete interval ID
Z such that [a, b]Z ⊆ ID
Z ⊆ IZ. Indeed, let us denote by
z = (x, u)⊤ ∈ C(I+
Z )2 a sequence satisfying system (6.22) with λ = 0 and assume that
ψk zk = 0 on [a, b]Z, i.e., the pair xk, uk is such that
xk =
uk+1
pk+1
, uk = qk xk+1 −
qk
pk+1
uk+1, and wk xk = 0 for all k ∈ [a, b]Z. (6.23)
Then the positivity of wk on [a, b]Z and the third equality in (6.23) yield xk = 0 for all
k ∈ [a, b]Z, which implies that also uk = 0 for all k ∈ [a + 1, b]Z by the first equality in (6.23).
Hence zk = 0 on [a + 1, b]Z and from the invertibility of the coefficient matrix in (6.22) we
easily obtain z ≡ 0 on [a, b]Z. It means that z is only the trivial solution of system (6.22)
with λ = 0 and the definiteness of system (6.22) on [a, b]Z follows by Lemma 6.2.3.
In addition, we point out that system (6.22) corresponds to the second order Sturm–
Liouville difference equation
− [pk yk−1(λ)] + qk yk(λ) = λwk yk(λ), k ∈ IZ. (6.24)
with p ∈ C(I+
Z ) and q, w ∈ C(IZ). More precisely, let y(λ) ∈ C(I+
Z ∪ {−1}) be a solution of
equation (6.24) on the discrete interval IZ. Then the pair xk(λ) := yk(λ), k ∈ I+
Z ∪ {−1},
and uk(λ) := pk yk−1(λ), k ∈ I+
Z , satisfies system (6.22) for all k ∈ IZ, compare with the
system corresponding to equation (2.8) in the case m = n = 1 and see also [4, Example 3.8].
On the other hand, if p0 0 and the pair x(λ), u(λ) ∈ C(I+
Z ) solves system (6.22), then
y(λ) ∈ C(I+
Z ∪ {−1}) defined as
yk(λ) :=



xk(λ), k ∈ I+
Z ,
x0(λ) − u0(λ)/p0, k = −1,
satisfies equation (6.24) for all k ∈ IZ. ▲
System (6.22) is a particular case of system (Sλ) with a special linear dependence on
the parameter λ, which is analogous to system (Sλ) with (2.7). Namely, let
Sk =
(
Ak Bk
Ck Dk
)
, Vk =
(
0 0
Wk Ak Wk Ak
)
, and ψk =
(
Wk 0
0 0
)
, (6.25)
– 95 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
where A, B, C, D, W ∈ C(IZ)n×n are such that S∗
k
JSk = J and Wk = W∗
k
≥ 0 for all k ∈ IZ.
Then the conditions in (6.1) are satisfied on IZ and in the following theorem we derive
a sufficient condition for the definiteness of system (Sλ) with the coefficients as in (6.25).
For completeness, we note that the latter system is equivalent with the pair of equations
xk(λ) = Ak xk+1(λ) + Bk uk+1(λ), uk(λ) = Ck xk+1(λ) + Dk uk+1(λ) + λWk xk(λ). (6.26)
Theorem 6.2.5. Let us consider system (Sλ) with the coefficients specified in (6.25). If there exists
an index m ∈ IZ K{0} such that the matrices Bm−1, Wm−1, and Wm are invertible (in fact, Wm−1
and Wm are positive definite), then system (Sλ) is definite on the discrete interval [m − 1, m]Z, and
thus also on IZ.
Proof. Let λ ∈ C be fixed and z(λ) = (x⊤(λ), u⊤(λ))⊤ ∈ C(I+
Z )2n be a nontrivial solution of
system (Sλ) such that ψk zk(λ) = 0 on [m − 1, m]Z, i.e., Wm−1 xm−1(λ) = 0 and Wm xm(λ) = 0.
By Lemma 6.2.3 we have to show that zk(λ) = 0 on [m − 1, m]Z. From the invertibility of
Wm−1 and Wm we obtain xm−1(λ) = xm(λ) = 0. Hence
0 = xm−1(λ)
(6.26)
= Am−1 xm(λ) + Bm−1 um(λ) = Bm−1 um(λ),
which yields also um(λ) = 0 by the invertibility of Bm−1. It means zm(λ) = 0 and consequently
z(λ) ≡ 0 on I+
Z . ■
For the following result we fix k0 ∈ I+
Z and by Θ(λ) ∈ C(I+
Z )2n×2n we denote the
fundamental matrix of system (Sλ) satisfying Θk0
(λ) = I for any λ ∈ C. Hence Θ(λ)
satisfies equality (6.15) with s = k0 and thus also the relations in (6.16) for all k ∈ I+
Z .
The next result provides a characterization of the definiteness of system (Sλ) which is
analogous to that for system (2.5) established in [13, Proposition 2.11].
Lemma 6.2.6. System (Sλ) is definite on IZ if and only if there exists a finite discrete interval
ID
Z ⊆ IZ over which the system is definite.
Proof. If the discrete interval IZ is finite, the statement is trivial. Hence, let us consider
the case IZ = [0, ∞)Z. From Remark 6.2.2 we know that the definiteness of system (Sλ) on
a finite discrete interval ID
Z implies definiteness on the discrete interval [0, ∞)Z. Thus, it
remains to show the converse.
Assume that system (Sλ) is definite on [0, ∞)Z. In light of Lemma 6.2.3, we need only
to show the existence of a finite discrete interval ID
Z over which system (Sλ) is definite
for one value λ0 ∈ C. Thus, let λ0 ∈ C be fixed and for any finite discrete subinterval
IZ ⊂ [0, ∞)Z we define the set s(IZ) as
s(IZ) :=
{
ξ ∈ C2n
||ξ||2 = 1 and
∑
k∈I
ξ∗
Θ∗
k(λ0)ψk Θk(λ0)ξ = 0
}
,
where ||·||2 is the Euclidean norm on C2n. Then s(IZ) is compact and it holds s(IZ) ⊆ s(IZ)
whenever IZ ⊆ IZ. Moreover, let
{
I [m]
Z
}
m∈N
be a collection of nested finite discrete intervals
I [1]
Z ⊆ I [2]
Z ⊆ · · · ⊂ [0, ∞)Z such that
∪
m∈N I [m]
Z = [0, ∞)Z. Suppose that there exists a vector
ξ ∈ C2n with ||ξ||2 = 1 such that for every m ∈ N we have
∑
k∈I[m] ξ∗ Θ∗
k
(λ0)ψk Θk(λ0)ξ = 0.
Then also
∑∞
k=0 ξ∗ Θ∗
k
(λ0)ψk Θk(λ0)ξ = 0, which implies that Θk(λ0)ξ = 0 for all k ∈ [0, ∞)Z
by the definiteness of system (Sλ) on [0, ∞)Z, see Remark 6.2.2. But it means that ξ = 0,
which contradicts the assumption ||ξ||2 = 1. Consequently
∩
m∈N
s
(
I [m]
Z
)
= ∅
– 96 –
6.2. Definiteness condition
for the collection of nested compact sets s
(
I [1]
Z
)
⊇ s
(
I [2]
Z
)
⊇ · · · . Thus by the Cantor
intersection theorem, q.v. [7, Theorem 3.25], there exists m0 ∈ N such that s
(
I
[m0]
Z
)
= ∅,
which demonstrates the definiteness of system (Sλ) on the interval ID
Z = I
[m0]
Z ⊂ [0, ∞)Z. ■
Remark 6.2.7. From Lemmas 6.2.3 and 6.2.6 we conclude that the strong Atkinson condition
stated similarly as in Hypothesis 2.4.11 for system (Sλ) is equivalent to the definiteness
of system (Sλ) on [0, ∞)Z and it suffices to verify this condition only for one λ ∈ C. Note
also that [26, Assumption 2.2] requires satisfaction of inequality (6.18) on every nonempty
finite subinterval of [0, ∞)Z, which is significantly stronger than requiring the definiteness
of system (Sλ) on [0, ∞)Z as seen, e.g., when (Sλ) is definite on a finite discrete interval
[0, N]Z ⊂ [0, ∞)Z and ψk ≡ 0 for k ∈ [N + 1, ∞)Z.
Now we establish a basic result concerning the solvability of a boundary value problem
associated with system (Sλ), which will be utilized in the proof of Lemma 6.4.3. It
provides the symplectic counterpart of the original Naimark’s result known as the “Patching
lemma”, see [124, Lemma 2 in Section 17.3]. Analogous result for system (2.6) can be
found in [135, Lemma 3.3].
Lemma 6.2.8. Let system (Sλ) be definite on a finite discrete interval ID
Z ⊆ IZ and a finite discrete
interval IZ := [c, d]Z be given such that ID
Z ⊆ IZ ⊆ IZ with c, d ∈ IZ. Then for any given
ξ, η ∈ C2n there exists f ∈ C(IZ)2n such that the boundary value problem
L (z)k = ψk fk, zc = ξ, zd+1 = η, k ∈ IZ, (6.27)
possesses a solution z ∈ C(I
+
Z )2n.
Proof. Let A = (aij)i,j=1,...,2n be the 2n × 2n matrix with the elements aij :=
∑d
k=c φ[i]∗
k
ψk φ
[j]
k
for i, j ∈ {1, . . . , 2n}, where φ[1]
, . . . , φ[2n]
∈ C(I+
Z )2n are linearly independent solutions of
system (S0), i.e., L (φ[i]
)k = 0 for all k ∈ IZ and i ∈ {1, . . . , 2n}. Then the homogeneous
system of algebraic equations Aρ = 0, where ρ = (ρ1, . . . , ρ2n)⊤ ∈ C2n, is equivalent to
∑d
k=c φ∗
k
ψk φk = 0, where φ :=
∑2n
i=1 ρi φ[i]
∈ C(I+
Z )2n. Since φ also solves system (S0), it
follows from the assumption of the definiteness on ID
Z and inequality (6.20) that φ is only
the trivial solution of system (S0), i.e.,
∑2n
i=1 ρi φ[i]
k
≡ 0, which implies that ρi = 0 for all
i ∈ {1, . . . , 2n}. Thus, the matrix A is invertible.
Consequently, there exists a unique solution ζ = (ζ1, . . . , ζ2n)⊤ ∈ C2n of the nonhomogeneous
system of algebraic equations
ζ∗
A = η∗
JΘd+1, (6.28)
where Θ := (φ[1]∗, . . . , φ[2n]∗)∗ is a fundamental matrix of system (S0). If we put h[1]
k
:= Θk ζ
for k ∈ IZ, we get from (6.28) for all i ∈ {1, . . . , 2n} that
d∑
k=c
h[1]∗
k
ψk φ[i]
k
= η∗
Jφ[i]
d+1
. (6.29)
Simultaneously the definiteness of system (Sλ) guarantees the existence of a unique
solution z[1]
∈ C(I
+
Z )2n of the nonhomogeneous initial value problem
L (z[1]
)k = ψk h[1]
k
, z[1]
c = 0, k ∈ IZ.
Then, for all i ∈ {1, . . . , 2n}, the fact L (φ[i]
)k ≡ 0 and identity (6.13) yield
d∑
k=c
h[1]∗
k
ψk φ[i]
k
=
d∑
k=c
{
L ∗
(z[1]
)k φ[i]
k
− z[1]∗
k
L (φ[i]
)k
}
= (z[1]
, φ[i]
)k
d+1
c
= (z[1]
, φ[i]
)d+1. (6.30)
– 97 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
Upon combining (6.29) and (6.30) we obtain z[1]
d+1
= η, which means that z[1]
solves the
boundary value problem
L (z[1]
)k = ψk h[1]
k
, z[1]
c = 0, z[1]
d+1
= η, k ∈ IZ.
Similarly, the nonhomogeneous system of algebraic equations ω∗A = ξ∗JΘc has a unique
solution ω = (ω1, . . . , ω2n)⊤ ∈ C2n. Then with h[2]
k
:= Θk ω, k ∈ IZ, we can calculate that
z[2]
∈ C(I
+
Z )2n, being the unique solution of
L (z[2]
)k = −ψk h[2]
k
, z[2]
d+1
= 0, k ∈ IZ,
also satisfies z[2]
c = ξ, i.e., it solves the boundary value problem
L (z[2]
)k = −ψk h[2]
k
, z[2]
c = ξ, z[2]
d+1
= 0, k ∈ IZ.
Therefore, the sequence z ∈ C(I
+
Z )2n with the terms zk := z[1]
k
+ z[2]
k
for all k ∈ I
+
Z solves the
boundary value problem (6.27) with fk := h[1]
k
− h[2]
k
for k ∈ IZ, i.e., f ∈ C(IZ)2n. ■
Finally, we derive yet another characterization of the definiteness of system (Sλ),
analogous to that given for linear Hamiltonian systems (2.5) and (2.6) with Ek ≡ 0 in [116,
Sections 2.3 and 2.4] and [134, Sections 3 and 4], respectively.
For any λ ∈ C and nonempty finite discrete interval IZ ⊆ IZ with k0 ∈ I
+
Z we define
the 2n × 2n positive semidefinite matrix
ϑ(λ, IZ) :=
∑
k∈I
Θ∗
k(λ)ψk Θk(λ) (6.31)
in terms of the fundamental matrix Θ(λ) of system (Sλ) specified in the paragraph preceding
Lemma 6.2.6. While the matrix ϑ(λ, IZ) depends obviously on λ and IZ, we next
establish that its kernel and range do not, which implies also the independence of the
rank of ϑ(λ, IZ) on the value of λ.
Lemma 6.2.9. For any nonempty finite discrete interval IZ ⊆ IZ such that k0 ∈ I
+
Z , the subspaces
Ker ϑ(λ, IZ) and Ran ϑ(λ, IZ) are independent of λ ∈ C.
Proof. Let λ ∈ C and ξ ∈ Ker ϑ(λ, IZ) be fixed and put zk := Θk(λ)ξ for all k ∈ I
+
Z . Then
obviously z ∈ C(I
+
Z )2n solves system (Sλ) on IZ, i.e., L (z) = λψz on IZ, while satisfying
the initial condition zk0
= ξ. Simultaneously, the sequence z solves also system (Sν) on IZ
for any ν ∈ C, i.e., L (z) = νψz on IZ, because the positive semidefiniteness of the elements
in the sum on the right-hand side of (6.31) implies ψk zk = ψk Θk(λ)ξ = 0 for all k ∈ IZ.
Hence zk = Θk(ν)ζ for all k ∈ I
+
Z and some ζ ∈ C2n. Since ξ = zk0
= Θk0
(ν)ζ = ζ, it holds
z = Θ(λ)ξ = Θ(ν)ξ on the discrete interval I
+
Z , which implies
0 = ξ∗
ϑ(λ, IZ)ξ =
∑
k∈I
z∗
k ψk zk = ξ∗
ϑ(ν, IZ)ξ.
Therefore, Ker ϑ(λ, IZ) ⊆ Ker ϑ(ν, IZ) and by reversing the roles of the numbers λ and ν
we obtain Ker ϑ(λ, IZ) = Ker ϑ(ν, IZ) for all λ, ν ∈ C. The independence of Ran ϑ(λ, IZ)
on λ ∈ C follows from the previous part and the fact that, as defined in (6.31), the matrix
ϑ(λ, IZ) is Hermitian, which yields Ran ϑ(λ, IZ) = Ker ϑ(λ, IZ)⊥ by (1.6). ■
– 98 –
6.2. Definiteness condition
The latter statement justifies the suppression of the parameter λ in the notation of the
kernel, range, and rank of ϑ(λ, IZ) for any nonempty finite discrete interval IZ ⊆ IZ with
k0 ∈ I
+
Z , i.e., henceforward we write only Ker ϑ(IZ), Ran ϑ(IZ), and rank ϑ(IZ). In the
following theorem we show that there exists a nonempty finite discrete interval IZ ⊆ IZ,
which maximizes the value of rank ϑ(·).
Lemma 6.2.10. There exists a nonempty finite discrete interval IZ ⊆ IZ with k0 ∈ I
+
Z such that
for any finite discrete interval IZ satisfying IZ ⊆ IZ ⊆ IZ we have
rank ϑ(IZ) = rank ϑ(IZ), Ran ϑ(IZ) = Ran ϑ(IZ). (6.32)
Proof. If the discrete interval IZ is finite, the statement is trivial. Hence, let us consider the
case IZ = [0, ∞)Z. For finite discrete intervals IZ and IZ such that k0 ∈ I
+
Z ⊆ I
+
Z ⊂ [0, ∞)Z,
we see that Ker ϑ(IZ) ⊆ Ker ϑ(IZ) by the definition of ϑ(·) in (6.31). Then, given that the
matrix ϑ(·) is Hermitian for any finite discrete interval, we see that
Ran ϑ(IZ)
(1.6)
= [Ker ϑ(IZ)]⊥
⊆ [Ker ϑ(IZ)]⊥ (1.6)
= Ran ϑ(IZ),
and consequently rank ϑ(IZ) ≤ rank ϑ(IZ) by (1.1), i.e., the value of rank ϑ(·) does not
decrease when we extend the interval. Since at the same time rank ϑ(·) ≤ 2n, there
must be a finite discrete interval IZ such that rank ϑ(IZ) = rank ϑ(IZ), and thus also
Ran ϑ(IZ) = Ran ϑ(IZ), for all finite discrete intervals IZ containing IZ. ■
Finally, we describe a connection between the definiteness of system (Sλ) and the
matrix ϑ(λ, IZ) for a finite discrete interval IZ ⊆ IZ.
Theorem 6.2.11. For a nonempty finite discrete interval IZ ⊆ IZ with k0 ∈ I
+
Z and for ϑ(λ, IZ)
being defined as in (6.31) the following statements are equivalent.
(i) It holds rank ϑ(IZ) = 2n.
(ii) It holds Ker ϑ(IZ) = {0}.
(iii) For some λ ∈ C, every nontrivial solution z(λ) ∈ C(I+
Z )2n of system (Sλ) satisfies∑
k∈I z∗
k
(λ)ψk zk(λ) > 0.
(iv) For some λ ∈ C, a solution z(λ) ∈ C(I+
Z )2n of system (Sλ) is necessarily trivial, i.e. zk(λ) = 0
for all k ∈ I+
Z , when
∑
k∈I z∗
k
(λ)ψk zk(λ) = 0.
Proof. The equivalence of (i) and (ii) is clear, while the equivalence of (iii) and (iv) follows
from Remark 6.2.2 and Lemma 6.2.3. Hence it remains to show that the statements in (ii)
and (iii) are equivalent.
Let us assume that (ii) is true and z(λ) ∈ C(I+
Z )2n be an arbitrary nontrivial solution of
system (Sλ) for some λ ∈ C, i.e., zk(λ) = Θk(λ)ξ for some ξ ∈ C2nK{0} and all k ∈ I+
Z . Since
ϑ(λ, IZ) is positive definite and ξ 0, we obtain
0 < ξ∗
ϑ(λ, IZ)ξ =
∑
k∈I
ξ∗
Θ∗
k(λ)ψk Θk(λ)ξ =
∑
k∈I
z∗
k(λ)ψk zk(λ),
i.e., (iii) holds. Conversely, assume that the statement in (iii) is true. Let ξ ∈ C2nK{0} be
fixed and put zk(λ) := Θk(λ)ξ for all k ∈ I+
Z . Then z(λ) ∈ C(I+
Z )2n is a nontrivial solution
of system (Sλ) and, by (iii),
ξ∗
ϑ(λ, IZ)ξ =
∑
k∈I
z∗
k(λ)ψk zk(λ) > 0,
– 99 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
i.e., we have ϑ(λ, IZ)ξ 0. Since the vector ξ was chosen arbitrarily, we conclude that
Ker ϑ(IZ) = {0}, i.e., (ii) is satisfied, which completes the proof. ■
As an immediate consequence of Lemma 6.2.6 and Theorem 6.2.11 we get the following
corollary; cf. [116, Definition 2.14].
Corollary 6.2.12. System (Sλ) is definite on IZ if and only if, for some nonempty finite discrete
interval IZ ⊆ IZ, one of the conditions listed in Theorem 6.2.11 is satisfied.
6.3 Nonhomogeneous problem
Now we take the nonhomogeneous problem into consideration and extend the results
of [A4, Section 5] to the case of general linear dependence on λ. For this purpose we
naturally consider only the case IZ = [0, ∞)Z and we start this section by restating some
fundamental results from the Weyl–Titchmarsh theory for system (Sλ), which are related
to the present study of system (S
f
λ
). As discussed at the beginning of this chapter, see (6.3),
these results can be easily derived from Chapter 2 by appropriate changes in the definition
of the semi-inner product and its weight matrix.
Throughout this section we assume that system (Sλ) is definite on [0, ∞)Z and “fix”
the fundamental matrix Θ(λ) ∈ C([0, ∞)Z)2n×2n by the initial condition Θ0(λ) = (α∗, −Jα∗)
for a given α ∈ , see (2.19) and (2.21). Since α ∈ , this fundamental matrix satisfies
equality (6.15) with s = 0, and thus all the relations in (6.16) hold. We also denote by
Z(λ), Z(λ) ∈ C([0, ∞)Z)2n×n the two components of the fundamental matrix, i.e., we put
Θ(λ) := (Z(λ), Z(λ)). If λ ∈ CKR, then the associated Weyl disks are defined in the same
way as in Definition 2.3.1 with the E(M)-function replaced by E(M) := iδ(λ)X∗(λ)JX(λ),
where M ∈ Cn×n and
Xk(λ) := Θk(λ, α)(I, M∗
)∗
, k ∈ [0, ∞)Z, (6.33)
represents the Weyl solution of system (Sλ); cf. identity (2.32) and Definition 2.2.1. Since
system (Sλ) is assumed to be definite on the discrete interval [0, ∞)Z, the limiting Weyl disk
exists and it is a closed, convex, and nonempty subset of Cn×n; cf. Definition 2.3.10 and
Remark 6.2.7. Hence the columns of the Weyl solution X(λ) defined through any matrix
M from the limiting Weyl disk are linearly independent square summable solutions
of system(Sλ), i.e., they belong to l2
ψ
; cf. Theorem 2.4.1. Consequently we adopt the
terminology from Definition 2.4.2, i.e., system (Sλ) is said to be in the limit point case
and in the limit circle case if it possesses n and 2n linearly independent solutions in l2
ψ
,
respectively. Finally, by M+(λ) we denote the half-line Weyl–Titchmarsh M(λ)-function,
which is defined in accordance with Remark 2.3.17(i) and satisfies
M∗
+(λ) = M+(¯λ) for all λ ∈ CKR. (6.34)
Moreover, if λ, ν ∈ CKR and systems (Sλ) and (Sν), are both in the limit point or in the
limit circle case, then
lim
k→∞
X+∗
k (λ)JX+
k (ν) = 0, (6.35)
whereX+(λ),X+(ν) ∈ C([0, ∞)Z)2n×n represent the Weyl solutions of systems (Sλ) and (Sν)
defined by (6.33) through the matrices M+(λ) and M+(ν), respectively; cf. Theorem 2.4.12.
For simplicity, we putX+∗
k
(λ) := [X+
k
(λ)]∗. The next result shows a useful relation between
the Weyl solution X+(λ) and Z(λ).
– 100 –
6.3. Nonhomogeneous problem
Lemma 6.3.1. Let α ∈ , λ ∈ CKR, and system (Sλ) be definite on [0, ∞)Z. Then
X+
k (λ)Z
∗
k(¯λ) − Zk(λ)X+∗
k (¯λ) = J for all k ∈ [0, ∞)Z. (6.36)
Proof. Identity (6.36) then follows by a direct calculation from the definition of X+(·), the
third identity in (6.16), and equality (6.34). ■
For λ ∈ CKR and k, s ∈ [0, ∞)Z we introduce the Green function
Gk,s(λ) :=



Zk(λ)X+∗
s (¯λ), k ∈ [0, s]Z,
X+
k
(λ)Z
∗
s(¯λ), k ∈ [s + 1, ∞)Z,
(6.37)
which can be equivalently expressed as
Gk,s(λ) =



X+
k
(λ)Z
∗
s(¯λ), s ∈ [0, k − 1]Z,
Zk(λ)X+∗
s (¯λ), s ∈ [k, ∞)Z.
(6.38)
Let us note that in the literature we also find the terminology resolvent kernel for an
analogous function in the continuous time case, see e.g. [107, page 15].
In the following lemma, we establish some fundamental properties of the Green
function. We note that the given identities are presented in a more symmetric form, with
respect to the variables k and s, than the corresponding identities for the Green function in
the case of the system (Sλ) with the special linear dependence on the spectral parameter
given in [A4, Lemma 5.1].
Lemma 6.3.2. Let α ∈ , λ ∈ CKR, and system (Sλ) be definite on [0, ∞)Z. Then the Green
function G(λ) ∈ C([0, ∞)Z)2n×2n possesses the following properties:
(i) G∗
k,s
(λ) = Gs,k(¯λ) for all k, s ∈ [0, ∞)Z such that k s;
(ii) G∗
k,k
(λ) = Gk,k(¯λ) + J for all k ∈ [0, ∞)Z;
(iii) for any given s ∈ [0, ∞)Z the function G·,s(λ) satisfies the homogeneous system (Sλ) for all
k ∈ [0, ∞)Z such that k s, i.e., Gk,s(λ) = pk(λ)Gk+1,s(λ) on the set
{
{k, s} ∈ [0, ∞)Z × [0, ∞)Z | k s
}
;
(iv) Gk,k(λ) = pk(λ)Gk+1,k(λ) − J for every k ∈ [0, ∞)Z;
(v) the columns of G·,s(λ) belong to l2
ψ
for every s ∈ [0, ∞)Z and the columns of Gk,·(λ) belong
to l2
ψ
for every k ∈ [0, ∞)Z.
Proof. The first property follows directly from the definition of Gk,s(λ) given in (6.37).
The second property can be obtained from (6.37) by means of identity (6.36). For the
proof of the third property we distinguish two cases for the calculation of the value of
Gk,s(λ) − pk(λ)Gk+1,s(λ) and use the fact that X+(λ) and Z(λ) solve system (Sλ), i.e.,
Gk,s(λ) − pk(λ)Gk+1,s(λ) =
{
[X+
k (λ) − pk(λ)X+
k+1(λ)]Z
∗
s(¯λ) = 0 when s < k < k + 1,
[Zk(λ) − pk(λ)Zk+1(λ)]X+∗
s (¯λ) = 0 when s ≥ k + 1 > k.
Property (iv) can be proven by using the definition of G(λ) in (6.37) together with identities
(6.34) and (6.16). Finally, the columns of G·,s(λ) belong to l2
ψ
for every s ∈ [0, ∞)Z
– 101 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
by the definition of the Green function and the square summability of the Weyl solution
X+(λ), because
||G·,s(λ)ej ||2
ψ =e∗
jX+
s (¯λ)
( s∑
k=0
Z
∗
k(λ)ψk Zk(λ)
)
X+∗
s (¯λ)ej
+ e∗
j Zs(¯λ)
( ∞∑
k=s+1
X∗+
k (λ)ψkX+
k (λ)
)
Z
∗
s(¯λ)ej < ∞,
while the columns of G∗
k,·
(λ) are in l2
ψ
for every k ∈ [0, ∞)Z by the similar calculation and
the equivalent expression given in (6.38). ■
Let us associate with system (S
f
λ
) the sequence z(λ) ∈ C([0, ∞)Z)2n, where
zk(λ) :=
∞∑
s=0
Gk,s(λ)ψs fs, k ∈ [0, ∞)Z. (6.39)
By (6.38), it can be written as
zk(λ) = X+
k (λ)
k−1∑
s=0
Z
∗
s(¯λ)ψs fs + Zk(λ)
∞∑
s=k
X+∗
s (¯λ)ψs fs, (6.40)
which shows that z(λ) is well defined for all f ∈ l2
ψ
by the Cauchy–Schwarz inequality,
because the columns of X+(¯λ) are square summable, see also Lemma 6.3.2(v). Similarly
as in [A4, Theorem 5.2] we show that the above defined function z(λ) represents a square
summable solution of system (S
f
λ
) with f ∈ l2
ψ
.
Theorem 6.3.3. Let α ∈ , λ ∈ CKR, f ∈ l2
ψ
, and system (Sλ) be definite on [0, ∞)Z. The sequence
z(λ) defined in (6.39) solves system (S
f
λ
) on [0, ∞)Z, satisfies the initial condition αz0(λ) = 0, is
square summable, i.e., z(λ) ∈ l2
ψ
, and it holds
||z(λ)||ψ ≤
1
| im(λ)|
|| f ||ψ. (6.41)
In addition, if system (Sλ) is in the limit point or limit circle case for all λ ∈ CKR, then
lim
k→∞
X+∗
k (ν)Jzk(λ) = 0 for every ν ∈ CKR. (6.42)
Proof. The form of zk(λ) given in (6.40) together with the similar expression of zk+1(λ),
the fact that X+(λ) and Z(λ) solve system (Sλ), and identity (6.36) yield
zk(λ) − pk(λ)zk+1(λ) = X+
k (λ)
k−1∑
s=0
Z
∗
s(¯λ)ψs fs + Zk(λ)
∞∑
s=k
X+∗
s (¯λ)ψs fs
− pk(λ)X+
k+1(λ)
k∑
s=0
Z
∗
s(¯λ)ψs fs − pk(λ)Zk+1(λ)
∞∑
s=k+1
X+∗
s (¯λ)ψs fs
= −
[
X+
k (λ)Z
∗
k(¯λ) − Zk(λ)X+∗
k (¯λ)
]
ψk fk = −Jψk fk,
i.e., the sequence z(λ) solves system (S
f
λ
).
– 102 –
6.3. Nonhomogeneous problem
The fulfillment of the boundary condition follows by the simple calculation
αz0(λ) = αZ0(λ)
∞∑
s=0
X+∗
s (¯λ)ψs fs = −αJα∗
∞∑
s=0
X+∗
s (¯λ)ψs fs = 0,
because Z0(λ) = −Jα∗ and α ∈ .
Next, we prove the estimate in (6.41) which together with the assumption f ∈ l2
ψ
will
imply that z(λ) ∈ l2
ψ
. For every r ∈ [0, ∞)Z we define the function
f[r]
k
:=



fk, k ∈ [0, r]Z,
0, k ∈ [r + 1, ∞)Z,
and the function
z[r]
k
(λ) :=
∞∑
s=0
Gk,s(λ)ψs f[r]
s =
r∑
s=0
Gk,s(λ)ψs fs. (6.43)
Then z[r]
(λ) solves system (S
f
λ
) with f replaced by f[r]
. Applying the extended Lagrange
identity from Theorem 6.1.3, we obtain
lim
k→∞
z[r]∗
k+1
(λ)Jz[r]
k+1
(λ) = z[r]∗
0
(λ)Jz[r]
0
(λ) + (¯λ − λ)
∞∑
k=0
z[r]∗
k
(λ)ψk z[r]
k
(λ)
+
∞∑
k=0
f[r]∗
k
ψk z[r]
k
(λ) −
∞∑
k=0
z[r]∗
k
(λ)ψk f[r]
k
.
(6.44)
Since Z0(λ) = −Jα∗ and α ∈ , we see that
z[r]∗
0
(λ)Jz[r]
0
(λ) =
( r∑
s=0
X+∗
s (¯λ)ψs fs
)∗
Z
∗
0(λ)JZ0(λ)
( r∑
s=0
X+∗
s (¯λ)ψs fs
)
= 0. (6.45)
For every k ∈ [r + 1, ∞)Z we can also write
z[r]
k
(λ) = X+
k (λ)gr(λ), where gr(λ) :=
r∑
s=0
Z
∗
s(¯λ)ψs fs, (6.46)
which, together with the fact that M+(λ) belongs to the limiting Weyl disk, yields
1
¯λ − λ
lim
k→∞
z[r]∗
k+1
(λ)Jz[r]
k+1
(λ) =
iδ(λ)
2| im(λ)|
g∗
r(λ)
(
lim
k→∞
X+∗
k+1(λ)JX+
k+1(λ)
)
gr(λ)
=
1
2| im(λ)|
g∗
r(λ)
(
lim
k→∞
Ek+1(M+(λ))
)
gr(λ) ≤ 0. (6.47)
By using identities (6.44), (6.45), and (6.47), the assumption λ ¯λ, the Hermitian property
of ψk, and the Cauchy–Schwarz inequality we get
||z[r]
(λ)||2
ψ =
∞∑
k=0
z[r]∗
k
(λ)ψk z[r]
k
(λ) ≤
1
2i im(λ)
( r∑
k=0
f[r]∗
k
ψk z[r]
k
(λ) −
r∑
k=0
z[r]∗
k
(λ)ψk f[r]
k
)
≤
1
| im(λ)|
r∑
k=0
z[r]∗
k
(λ)ψk f[r]
k
≤
1
| im(λ)|
( r∑
k=0
z[r]∗
k
(λ)ψk z[r]
k
(λ)
)1/2 ( r∑
k=0
f[r]∗
k
ψk f[r]
k
)1/2
≤
1
| im(λ)|
||z[r]
(λ)||ψ × || f[r]
||ψ,
– 103 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
thereby yielding the inequality
||z[r]
(λ)||ψ ≤
1
| im(λ)|
|| f[r]
||ψ ≤
1
| im(λ)|
|| f ||ψ, (6.48)
because z[r]
(λ) ∈ l2
ψ
. Upon combining identities (6.39) and (6.43) for any k, r ∈ [0, ∞)Z we
easily calculate
zk(λ) − z[r]
k
(λ) =
∞∑
s=r+1
Gk,s(λ)ψs fs.
Now, let t ∈ [0, r]Z. Then from the definition of G(λ) given in (6.37) we obtain for every
k ∈ [0, t]Z that
zk(λ) − z[r]
k
(λ) = Zk(λ)
∞∑
s=r+1
X+∗
s (¯λ)ψs fs. (6.49)
Since the columns of the Weyl solution X+(¯λ) and the function f belong to l2
ψ
, it follows
that
∑∞
s=0 X+∗
s (¯λ)ψs fs < ∞ by the Cauchy–Schwarz inequality. Hence the right-hand side
of (6.49) tends to zero as r → ∞ for every k ∈ [0, t]Z, which shows that z[r]
converges
uniformly to z(λ) on the interval [0, t]Z. Moreover, by (6.48), we have
t∑
k=0
z[r]∗
k
(λ)ψk z[r]
k
(λ) ≤ ||z[r]
(λ)||2
ψ ≤
1
| im(λ)|2
|| f ||2
ψ,
from which, as a consequence of the uniform convergence of z[r]
(λ) for r → ∞ on [0, t]Z,
we see that
t∑
k=0
z∗
k(λ)ψk zk(λ) ≤
1
| im(λ)|2
|| f ||2
ψ. (6.50)
As identity (6.50) is satisfied for any t ∈ [0, ∞)Z, the desired estimate in (6.41) follows.
Finally, to establish the existence of the limit in (6.42), assume that system (Sλ) is in the
limit point case for all λ ∈ CKR and ν ∈ CKR is arbitrary. From the extended Lagrange
identity in Theorem 6.1.3, for any k, r ∈ [0, ∞)Z, we obtain
[
X+∗
j (ν)Jz[r]
j
(λ)
]k+1
0
= (¯ν − λ)
k∑
j=0
X+∗
j (ν)ψj z[r]
j
(λ) −
k∑
j=0
X+∗
j (ν)ψj f[r]
j
(λ). (6.51)
If r ∈ [0, ∞)Z and k ∈ [r + 1, ∞)Z, then identities (6.35) and (6.46) imply
lim
k→∞
X+∗
k+1(ν)Jz[r]
k+1
(λ)
(6.46)
= lim
k→∞
X+∗
k+1(ν)JX+
k+1(λ)gr(λ)
(6.35)
= 0.
Hence upon taking in (6.51) the limit for k → ∞ we get
X+∗
0 (ν)Jz[r]
0
(λ) = (λ − ¯ν)
∞∑
j=0
X+∗
j (ν)ψj z[r]
j
(λ) +
∞∑
j=0
X+∗
j (ν)ψj f[r]
j
(λ). (6.52)
Since, by the previous part, z[r]
(λ) converges uniformly on finite subintervals of [0, ∞)Z to
z(λ) as r → ∞, identity (6.52) yields
X+∗
0 (ν)Jz0(λ) = (λ − ¯ν)
∞∑
j=0
X+∗
j (ν)ψj zj(λ) +
∞∑
j=0
X+∗
j (ν)ψj fj(λ). (6.53)
– 104 –
6.3. Nonhomogeneous problem
Simultaneously, as in (6.51), we obtain from (6.11) for every k ∈ [0, ∞)Z that
[
X+∗
j (ν)Jzj(λ)
]k+1
0
= (¯ν − λ)
k∑
j=0
X+∗
j (ν)ψj zj(λ) −
k∑
j=0
X+∗
j (ν)ψj fj(λ). (6.54)
Hence by letting k → ∞ in (6.54) and using equality (6.53) the limit in (6.42) is established.
An argument, similar to that given above, can be used also in the limit circle case to
show the existence of the limit in (6.42), because all solutions of system (Sλ) are square
summable in that case. However, an alternative and more direct method of the proof is
available. It utilizes the fact that Z(¯λ) ∈ l2
ψ
in the limit circle case, which is not true in the
limit point case, q.v. the proof of Theorem 2.4.3. More specifically, by (6.40) we have for
every k ∈ [0, ∞)Z that
X+∗
k (ν)Jzk(λ) = X+∗
k (ν)JX+
k (λ)
k−1∑
s=0
Z
∗
s(¯λ)ψs fs + X+∗
k (ν)JZk(λ)
∞∑
s=k
X+∗
s (¯λ)ψs fs. (6.55)
The limit in (6.42) follows by the fact that both the terms on the right-hand side of (6.55)
tend to zero as k → ∞. Indeed, the zero limit of the first term is a consequence of the
relation in (6.35) together with the convergence of the sum for k → ∞, which we get by
the Cauchy–Schwarz inequality. The second term tends to zero because X+∗(ν)JZ(λ) is
bounded by equality (6.11) and the sum converges to zero as k → ∞. ■
In the last result of this section, we extend [A4, Corollary 5.3] to the case of general
linear dependence on the spectral parameter.
Corollary 6.3.4. Let α ∈ , λ ∈ CKR, f ∈ l2
ψ
, and ξ ∈ Cn. Assume that system (Sλ) is definite
on the discrete interval [0, ∞)Z, and define
ˆzk(λ) := X+
k (λ)ξ + zk(λ), k ∈ [0, ∞)Z, (6.56)
where zk(λ) is given in (6.39). Then the sequence ˆz(λ) ∈ C([0, ∞)Z)2n represents a square
summable solution of system (S
f
λ
) satisfying the initial condition α ˆz0(λ) = ξ and
|| ˆz(λ)||ψ ≤
1
| im(λ)|
|| f ||ψ + ||X+
(λ)ξ||ψ. (6.57)
If system (Sλ) is in the limit point or in the limit circle case for all λ ∈ CKR, we have
lim
k→∞
X+∗
k (ν)J ˆzk(λ) = 0 for every ν ∈ CKR. (6.58)
Moreover, in the limit point case the sequence ˆz(λ) is the unique square summable solution of
system (S
f
λ
) satisfying α ˆz0(λ) = ξ, while in the limit circle case ˆz(λ) is the unique solution of (S
f
λ
)
being in l2
ψ
such that α ˆz0(λ) = ξ and
lim
k→∞
X+∗
k (¯λ)J ˆzk(λ) = 0. (6.59)
Proof. Since X+(λ)ξ solves system (Sλ) and Θ0(λ, α) = (α∗, −Jα∗), it follows from Theorem
6.3.3 that the sequence ˆz(λ) solves the nonhomogeneous system (S
f
λ
) and satisfies
α ˆz0(λ) = αX+
0
(λ)ξ = ξ. The estimate in (6.57) follows directly from (6.56) and (6.41) by
– 105 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
the triangle inequality. The limit in (6.58) follows in the limit point or in the limit circle
case from (6.35), (6.42), and from the calculation
lim
k→∞
X+∗
k (ν)J ˆzk(λ) = lim
k→∞
{
X+∗
k (ν)JX+
k (λ)ξ + X+∗
k (ν)Jzk(λ)
}
= 0.
Finally, we prove the uniqueness of the solution in the limit point and limit circle cases.
Assume that z[1]
(λ), z[2]
(λ) ∈ C([0, ∞)Z)2n are two square summable solutions of system (S
f
λ
)
satisfying αz[1]
0
(λ) = ξ = αz[2]
0
(λ). Then zk(λ) := z[1]
k
(λ) − z[2]
k
(λ), k ∈ [0, ∞)Z, represents
a square summable solution of the homogeneous system (Sλ) and it satisfies αz0(λ) = 0.
Since z(λ) = Θ(λ, α)ζ for some ζ ∈ C2n, the initial condition αz0(λ) = 0 implies that
zk(λ) = Zk(λ)η for some η ∈ Cn. If system (Sλ) is in the limit point case, we have z(λ) l2
ψ
for η 0, because the columns of Z(λ) do not belong to l2
ψ
in this case, see Theorem 2.4.3.
Therefore η = 0 and the uniqueness follows. On the other hand, if (Sλ) is in the limit circle
case and both z[1]
(λ) and z[2]
(λ) satisfy also the limit relation given in (6.59), then we obtain
from the previous part and the first identity in (6.16) that
0 = lim
k→∞
X+∗
k (¯λ)Jzk(λ) = lim
k→∞
X+∗
k (¯λ)JΘk(λ)
(
0
η
)
= lim
k→∞
(
I M∗
+(¯λ)
)
Θ∗
k(¯λ)JΘk(λ)
(
0
η
)
= η,
which implies the uniqueness of the solution ˆz(λ) also in the latter case. ■
6.4 Maximal and minimal linear relations
Finally, in this section we come to the topic of linear relations. We focus on a pair of linear
relations defined in terms of the linear map, L (·), introduced in (6.5) in association with
system (Sλ). Let us mention that similar results for linear Hamiltonian differential and
difference systems can be found in [13,116,134]. Moreover, we remind that a short introduction
to the theory of linear relations is available in the Appendix and we recommend
to read this passage now if it is not familiar to the reader.
For the present treatment we need to introduce some spaces in addition to the space
of square summable sequences l2
ψ
defined in (6.9). Namely, we denote by ˜l2
ψ
the quotient
space obtained by factoring out the kernel of the semi-norm ||·||ψ, i.e.,
˜l2
ψ := l2
ψ
/{
z ∈ C(I+
Z )2n
| ||z||ψ = 0
}
. (6.60)
It is easy to see that ˜l2
ψ
is a Banach space with respect to the norm generated by the
quotient space map π(z) := ˜z and simultaneously it is a Hilbert space with respect to the
associated inner product ⟨˜z, ˜v⟩ψ := ⟨z, v⟩ψ, where z and f are elements of the equivalence
classes ˜z ∈ ˜l2
ψ
and ˜f ∈ ˜l2
ψ
, respectively; cf. [142, Lemma 2.5]. Henceforward, we denote
by the superscript ˜· the corresponding equivalence class, i.e., if z ∈ l2
ψ
then ˜z ∈ ˜l2
ψ
is such
that z ∈ ˜z. In addition, we point out that the value of ||z||ψ :=
√
⟨z, z⟩ψ for z ∈ C(I+
Z )2n
does not depend on zN+1 in the case of IZ being a finite discrete interval, which implies
that the sequences z, v ∈ C(I+
Z )2n such that zk vk only for k = N + 1, belong to the same
equivalence class. In addition, the product space ˜l2×2
ψ
:= ˜l2
ψ
× ˜l2
ψ
is also a pre-symplectic
space with the associated function [· : ·] : ˜l2×2
ψ
× ˜l2×2
ψ
→ C given by
[{˜z, ˜f} : {w, ˜g}] := ⟨ ˜f, w⟩ − ⟨˜z, ˜g⟩ , (6.61)
– 106 –
6.4. Maximal and minimal linear relations
see (6.14) and the second part of the Appendix. Moreover, we define the subspace
l2
ψ,0 :=



{
z ∈ C0(I+
Z )2n
∩ l2
ψ | z0 = 0, zN+1 = 0
}
if IZ = [0, N]Z, N ∈ N ∪ {0},
{
z ∈ C0(I+
Z )2n
∩ l2
ψ | z0 = 0
}
if IZ = [0, ∞)Z,
and if IZ = [0, ∞)Z also
l2
ψ,1 :=
{
z ∈ l2
ψ | there exists K ∈ [0, ∞)Z such that ψk zk = 0 for all k ∈ [K, ∞)Z
}
.
Finally, the space ˜l2
ψ,1
consists of equivalence classes similarly as ˜l2
ψ
and it is easy to see
that ˜l2
ψ,1
is a dense subspace of ˜l2
ψ
.
6.4.1 Linear relations and definiteness
Now we can introduce the maximal linear relation Tmax as a subspace of ˜l2×2
ψ
given by
Tmax :=
{
{˜z, ˜f} ∈ ˜l2×2
ψ | there exists z ∈ ˜z such that L (z) = ψ f
}
. (6.62)
Note that when L (z) = ψ f, then L (z) = ψg for any g ∈ ˜f, i.e., the definition of Tmax does
not depend on the choice of the representative g ∈ ˜f. Similarly, we define the pre-minimal
linear relation
T0 :=
{
{˜z, ˜f} ∈ ˜l2×2
ψ | there exists z ∈ ˜z ∩ l2
ψ,0 such that L (z) = ψ f
}
, (6.63)
which evidently satisfies T0 ⊆ Tmax. In addition, by (A.6) we put
T0 − λI :=
{
{˜z, ˜f} ∈ ˜l2×2
ψ | there exists z ∈ ˜z ∩ l2
ψ,0 such that L (z) = λψz + ψf
}
. (6.64)
The consideration of linear relations (instead of operators) in our current context is
natural given that the weight ψ, being present on the right-hand side of system (S
f
λ
) and
in the definitions of the sequence spaces associated with Tmax and Tmin, has terms none
of which are positive definite, but all of which are only positive semidefinite, see (6.7).
Moreover, this fact is affirmed by the following simple example, which is analogous to
that found in [116, Section 2] for system (2.5).
Example 6.4.1. Let n = 1 and consider system (S
f
0
) with
Sk ≡
(
1 0
0 1
)
, ψk ≡
(
1 0
0 0
)
, fk =


f[1]
k
f[2]
k

 , and zk =
(
xk
uk
)
. (6.65)
Then L (z) and (S
f
0
), respectively, can be written as
L (z) =
(
0 −1
1 0
)
z and z =
(
x
u
)
=
(
0
−f[1]
)
. (6.66)
Hence, for any f ∈ l2
ψ
, the sequence z ∈ C(I+
Z )2 with the terms
z0 = 0 and zk =


0
−
∑k−1
j=0 f[1]
j

 for all k ∈ I+
Z K{0}
– 107 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
solves the nonhomogeneous system in (6.66), i.e., system (S
f
0
) with the coefficients specified
in (6.65). Since obviously z ∈ ˜0, we have {˜0, ˜f} ∈ Tmax for any ˜f ∈ ˜l2
ψ
. Thus, the
multivalued part of the corresponding linear relation Tmax, i.e., mul Tmax defined in (A.3),
is nontrivial, which means that Tmax is not a graph of a linear operator. Let us also note
that a solution of system (S
f
0
) in (6.66) which is an element of the space l2
ψ,0
, of necessity is
such that xk = 0 for all k ∈ I+
Z with the consequence that dom T0 = {˜0}, i.e., the set dom T0
is not dense in l2
ψ
. ▲
The latter example illustrates yet another interesting situation: for any c ∈ C the
sequence zk(λ) ≡ (0, c)⊤ ∈ ˜0 represents a solution of the nonhomogeneous system in (6.66)
with f ∈ ˜0, i.e., f[1]
k
≡ 0. It means that for the pair {˜0, ˜0} there exists infinitely many
representatives z ∈ ˜0 such that L (z) = ψf = 0 and, at the same time, it implies that
system (Sλ) with the coefficients given in (6.65) is not definite on the discrete interval IZ
by Lemma 6.2.3. In the next result we characterize the definiteness of system (Sλ) in terms
of the domain of Tmax.
Theorem 6.4.2. System (Sλ) is definite on the discrete interval IZ if and only if for any pair
{˜z, ˜f} ∈ Tmax there exists a unique z ∈ ˜z such that L (z) = ψ f.
Proof. Assume that system (Sλ) is definite on IZ. Let {˜z, ˜f} ∈ Tmax and z[1]
, z[2]
∈ ˜z be two
representatives of the equivalence class ˜z such that L (z[1]
) = ψ f = L (z[2]
). If we put
vk := z[1]
k
− z[2]
k
for all k ∈ I+
Z , then L (v) = 0 and v ∈ ˜0 ∈ ˜l2
ψ
, i.e., v solves system (S0)
and satisfies
∑
k∈I v∗
k
ψk vk = 0. Therefore, by Remark 6.2.2, the definiteness of system (Sλ)
implies that vk ≡ 0 on I+
Z , i.e., z[1]
≡ z[2]
on I+
Z and so there exists only one representative
z ∈ ˜z such that L (z) = ψf.
To show the converse, assume that there is only one z ∈ ˜z for which L (z) = ψf,
whenever {˜z, ˜f} ∈ Tmax. Let IZ ⊆ IZ be a finite discrete interval such that rank ϑ(IZ) is
maximal; cf. Lemma 6.2.10. If rank ϑ(IZ) < 2n, then there is a vector η ∈ C2nK{0} such
that η∗ ϑ(0, IZ)η =
∑
k∈I η∗ Θ∗
k
ψk Θk η = 0, where Θk := Θk(0) is the fundamental matrix
of system (Sλ) specified in the paragraph preceding Lemma 6.2.6 with k0 ∈ I
+
Z . If also∑
k∈I η∗Θ∗
k
ψkΘkη = 0, then v := Θη ∈ l2
ψ
, L (v) = 0, and v ∈ ˜0. Given that the zero
sequence is the unique representative of ˜0 satisfying L (z) = 0, it follows vk = Θk η = 0 for
all k ∈ I+
Z and as a consequence η = 0, which contradicts the assumption η 0. Hence,
there is a finite discrete interval superset IZ ⊃ IZ such that ϑ(0, IZ)η 0. As a result,
Ker ϑ(IZ) ⊂ Ker ϑ(IZ) and consequently Ran ϑ(IZ) ⊂ Ran ϑ(IZ); thereby contradicting the
maximality of rank ϑ(IZ). Thus, rank ϑ(IZ) = 2n and system (Sλ) is definite on the discrete
interval IZ by Theorem 6.2.11. ■
Naturally, the latter statement does not mean that the equivalence class ˜z contains only
one representative as it can be easily seen, e.g., from the example discussed in Remark 6.4.4
below. Nevertheless, if system (Sλ) is definite on the discrete interval IZ and {˜z, ˜f} ∈ Tmax,
then we denote the unique representative z ∈ ˜z satisfying L (z) = ψf by ˆz, i.e., ˆz := z. In
this case, identities (6.14) and (6.61) yield
[{˜z, ˜f} : {˜v, ˜g}] = ⟨ f, ˆv⟩ − ⟨ˆz, g⟩ = (ˆz, ˆv)k
N+1
0
(6.67)
for any {˜z, ˜f}, {˜v, ˜g} ∈ Tmax, where f ∈ ˜f and g ∈ ˜g are arbitrary representatives. Moreover,
we obtain from Lemma 6.2.8 the following statement, compare with [135, Remark 3.2]
and [147, Lemma 3.3].
– 108 –
6.4. Maximal and minimal linear relations
Lemma 6.4.3. Let system (Sλ) be definite on a finite discrete interval ID
Z := [a, b]Z ⊆ IZ, where
a, b ∈ IZ. Then for any pairs {˜z, ˜f}, {˜v, ˜g} ∈ Tmax there exists {˜r, ˜h} ∈ Tmax such that
ˆrk =



ˆzk, k ∈ [0, c]Z,
ˆvk, k ∈ [d + 1, ∞)Z ∩ I+
Z ,
where c ∈ [0, a]Z and d ∈ [b, ∞)Z ∩ IZ.
In particular, there exists {˜z[i]
, ˜f[i]
} ∈ Tmax such that ˆz[i]
0
= ei and ˆz[i]
k
= 0 for k ∈ [d+1, ∞)Z ∩I+
Z ,
where i ∈ {1, . . . , 2n} is arbitrary and ei = (0, . . . , 1, . . . , 0)⊤ ∈ C2n is the i-th canonical unit vector
in C2n. If, in addition, IZ is a finite discrete interval, i.e., IZ = [0, N]Z with N ∈ N ∪ {0}, then
there exists {˜r, ˜h} ∈ Tmax such that ˆrk = 0 for k ∈ [0, c]Z and ˆrN+1 = ei, where i ∈ {1, . . . , 2n} and
ei are the same as above.
Proof. Let IZ be a finite discrete interval as in Lemma 6.2.8, the pairs {˜z, ˜f}, {˜v, ˜g} ∈ Tmax
be arbitrary, and define ξ := ˆzc, η := ˆvd+1. Then by the latter lemma there exist sequences
ℓ ∈ C(IZ)2n and s ∈ C(I
+
Z )2n such that
L (s)k = ψk ℓk, sc = ξ, sd+1 = η, k ∈ IZ.
If we put
rk :=



ˆzk, k ∈ [0, c]Z ∩ IZ,
sk, k ∈ [c + 1, d]Z ∩ IZ,
ˆvk, k ∈ [d + 1, ∞)Z ∩ I+
Z
hk :=



fk, k ∈ [0, c − 1]Z ∩ IZ,
ℓk, k ∈ [c, d]Z ∩ IZ,
gk, k ∈ [d + 1, ∞)Z ∩ IZ,
then obviously r, h ∈ l2
ψ
and it can be verified by a direct calculation that L (r)k = ψk hk for
all k ∈ IZ, i.e., {˜r, ˜h} ∈ Tmax with ˆrk ≡ rk. The second part of the statement follows directly
from Lemma 6.2.8. ■
Remark 6.4.4. In Example 6.4.1 we constructed the linear map L and the nonhomogeneous
system, see (6.66), with two significant properties: (i) the maximal linear relation
does not determine a linear operator and (ii) the domain of the pre-minimal linear relation
is not is dense in ˜l2
ψ
. In addition, let us consider system (6.22) from Example 6.2.4 with
the coefficients pk ≡ 1, qk ≡ 0, and wk ≡ 1 on IZ with N 0, which imply the definiteness
of system (6.22) on IZ. If we put fk =
(
f[1]
k
, f[2]
k
)⊤
with f0 = (1, 0)⊤ and fk = (0, 0)⊤ for all
k ∈ IZ K{0}, then the corresponding nonhomogeneous system zk = Sk zk+1 − Jψk fk, i.e.,
(
xk
uk
)
=
(
1 −1
0 1
) (
xk+1
uk+1
)
+
(
0
f[1]
k
)
,
possesses the solution zk = (xk, uk)⊤, where z0 = (0, 1)⊤ and zk ≡ (0, 0)⊤ on I+
Z K{0}, i.e., there
exists ˜f ˜0 such that {˜0, ˜f} ∈ Tmax. This shows that even the definiteness of system (Sλ)
does not suffice to get mul Tmax = {˜0}; cf. [147].
Thus, to guarantee that the maximal linear relation defines an operator, we need to
assume explicitly that mul Tmax = {˜0}, i.e., if there exists z ∈ ˜0 such that L (z) = ψf for
some f ∈ l2
ψ
, then z ≡ 0; cf. [108, pg. 666]. In other words, we assume the “definiteness” of
system (S
f
0
) for every f ∈ l2
ψ
. Then system (Sλ) is definite as it follows by the choice f ≡ 0,
see Theorem 6.2.11(iii) and Corollary 6.2.12. Moreover, in the next theorem we prove that
this assumption even yields the density of dom T0 in ˜l2
ψ
; cf. [108, Theorem 7.6]. As noted
in [10, pg. 3], a similar condition is also needed for the study of operators associated with
system (2.6) in [142].
– 109 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
Theorem 6.4.5. If system (S
f
0
) is definite on IZ, then dom T0 is dense in ˜l2
ψ
.
Proof. Assume that dom T0 is not dense in ˜l2
ψ
, i.e., there exists ˜f ∈ (dom T0)⊥ such that
|| f ||ψ 0. Let ˜z ∈ dom T0 be such that L (z) = ψ f and ˜v ∈ dom T0 be such that L (v) = ψg
for some g ∈ l2
ψ
. Then, by identity (6.14), we obtain
⟨z, g⟩ψ = ⟨ f, v⟩ψ = 0, (6.68)
because z, v ∈ l2
ψ,0
. Since g ∈ l2
ψ
was chosen arbitrarily and z ∈ l2
ψ,0
, we can take g = z
and the solution of L (v) = ψz can be obtained as vk = Θk J
∑k−1
j=0 Θ∗
j
ψj zj, where Θ := Θ(0)
means a fundamental matrix of (S0) satisfying (6.15) for any s ∈ I+
Z . Then equality (6.68)
implies that ⟨z, z⟩ψ = 0, i.e., ψk zk = 0 on IZ. Thus we have L (z) = ψf and z ∈ ˜0, which
yields z ≡ 0 by the definiteness assumption for system (S
f
0
). So ψk fk = 0 on IZ, i.e., f ∈ ˜0,
and the density of dom T0 in ˜l2
ψ
is thus established. ■
6.4.2 Orthogonal decomposition of sequence spaces
In this subsection we introduce a linear map which will allow orthogonal decompositions
of ˜l2
ψ
and ˜l2
ψ,1
depending on the cardinality of the discrete interval IZ. In particular, let
us denote by Kλ the linear map defined by
Kλ :



˜l2
ψ → C2n
if IZ is a finite discrete interval,
˜l2
ψ,1 → C2n
if IZ = [0, ∞)Z,
Kλ(˜z) :=
∑
k∈I
Θ∗
k(¯λ)ψk zk, (6.69)
where Θ(λ) is the fundamental matrix of system (Sλ) specified in the paragraph preceding
Lemma 6.2.6 with k0 ∈ I+
Z . If λ = 0 we write only K(·) instead of K0(·). For completeness,
we note that the sum
∑
k∈I Θ∗
k
(¯λ)ψk zk does not depend on the choice of the representative
z ∈ ˜z ∈ ˜l2
ψ
or z ∈ ˜z ∈ ˜l2
ψ,1
, i.e., the map Kλ is defined correctly. In the following statement
we utilize the matrix ϑ(·) defined in (6.31).
Lemma 6.4.6. Let IZ be a finite discrete interval and λ ∈ C. Then Ran Kλ is independent of
λ ∈ C; in particular,
Ran Kλ =
{
ξ ∈ C2n
| ψk Θk(¯λ)ξ = 0 for all k ∈ IZ
}⊥
= Ran ϑ(IZ). (6.70)
Furthermore, the space ˜l2
ψ
admits the following orthogonal sum decomposition
˜l2
ψ = Ker Kλ ⊕
{
˜z ∈ ˜l2
ψ | z = Θ(¯λ)ξ, ξ ∈ Ran ϑ(IZ)
}
. (6.71)
Proof. For any ξ ∈ C2n and ˜v ∈ ˜l2
ψ
, we have
⟨ξ, Kλ(˜v)⟩C2n =
∑
k∈I
ξ∗
Θ∗
k(¯λ)ψk vk = ⟨Θ(¯λ)ξ, v⟩ψ.
Hence we have K∗
λ
: C2n → ˜l2
ψ
with K∗
λ
(ξ) := ˜z for ξ ∈ C2n, where ˜z is the equivalence
class corresponding to z = Θ(¯λ)ξ. In particular,
Ran K∗
λ =
{
˜z ∈ ˜l2
ψ | z = Θ(¯λ)ξ ∈ ˜z, ξ ∈ C2n
}
– 110 –
6.4. Maximal and minimal linear relations
and
Ker K∗
λ =
{
ξ ∈ C2n
| ||Θ(¯λ)ξ||ψ = 0
}
=
{
ξ ∈ C2n
| ψk Θk(¯λ)ξ = 0 for all k ∈ IZ
}
.
Thus the first equality in (6.70) follows from the fact that Ran Kλ = (Ker K∗
λ
)⊥.
Next, let us define the sequences z := Θ(¯λ)ξ ∈ C(I+
Z )2n, v := Θ(¯λ)η ∈ C(I+
Z )2n, and
r := Θ(¯λ)ζ ∈ C(I+
Z )2n, where ξ, η, ζ ∈ C2n are such that ξ = η + ζ with η ∈ Ran Kλ and
ζ ∈ (Ran Kλ)⊥ = Ker K∗
λ
. Then ||r||ψ = ||Θ(¯λ)ζ||ψ = 0, i.e., ˜r = ˜0 ∈ ˜l2
ψ
. Thus ˜z = ˜v and
Ran K∗
λ =
{
˜z ∈ ˜l2
ψ | z = Θ(¯λ)η, η ∈ Ran Kλ
}
.
So, to complete the demonstration of equalities (6.70) and (6.71), it remains to show that
Ran Kλ = Ran ϑ(IZ), because ˜l2
ψ
= Ker Kλ ⊕ (Ker Kλ)⊥ = Ker Kλ ⊕ Ran K∗
λ
.
Hence, let now ˜z ∈ ˜l2
ψ
. Then by the previous part ˜z = ˜r + ˜v, where ˜r ∈ Ker Kλ and the
equivalence class ˜v corresponds to v = Θ(¯λ)η with η ∈ Ran Kλ. Therefore
Kλ(˜z) = Kλ(˜v) =
∑
k∈I
Θ∗
k(¯λ)ψk Θk(¯λ)η = ϑ(λ, IZ)η,
which implies Ran Kλ ⊆ Ran ϑ(IZ). On the other hand, if ξ ∈ C2n and z = Θ(¯λ)ξ, then
ϑ(λ, IZ)ξ =
∑
k∈I
Θ∗
k(¯λ)ψk Θk(¯λ)ξ =
∑
k∈I
Θ∗
k(¯λ)ψk zk = Kλ(˜z);
thereby showing that Ran Kλ = Ran ϑ(IZ). ■
In the next two lemmas we establish similar results in the case IZ = [0, ∞)Z. Let
us recall that then there is a finite discrete interval IZ ⊂ [0, ∞)Z for which the value of
rank ϑ(IZ) is maximal by Lemma 6.2.10, i.e., rank ϑ(IZ) = rank ϑ(IZ) for any finite discrete
interval IZ such that IZ ⊆ IZ, viz. (6.32). Moreover, if IZ is a finite discrete interval for
which rank ϑ(IZ) is maximal, then we have Ran Kλ,I = Ran Kλ for any finite discrete
interval IZ such that IZ ⊆ IZ ⊂ [0, ∞)Z, see (6.70). Here Kλ,I means the map Kλ defined
as in (6.69) with IZ replaced by IZ. Especially, dom Kλ,I = ˜l2
ψ,I
, where ˜l2
ψ,I
is given as
in (6.60) for the discrete interval IZ instead of IZ.
Lemma 6.4.7. Let IZ = [0, ∞)Z and IZ ⊂ [0, ∞)Z be a finite discrete interval for which rank ϑ(IZ)
is maximal. Then Ran Kλ = Ran ϑ(IZ) for all λ ∈ C; in particular, Ran Kλ is independent of λ.
Proof. Let ˜z ∈ dom Kλ = ˜l2
ψ,1
. Then there exists K ∈ [0, ∞)Z such that ψk zk = 0 for all
k ∈ [K, ∞)Z. Hence, z ∈ l2
ψ
and ˜z ∈ ˜l2
ψ,[0,K]Z
= dom Kλ,[0,K], where ˜l2
ψ,[0,K]Z
is defined
similarly as ˜l2
ψ
in (6.60) with IZ = [0, K]Z. In particular, Kλ,[0,N](˜z) = Kλ(˜z), and thus
Ran Kλ ⊆ Ran Kλ,[0,N]. Without loss of generality, we may assume that IZ ⊆ [0, K]Z, and
hence that Ran Kλ,I = Ran Kλ,[0,N] ⊇ Ran Kλ.
Conversely, suppose that ˜z ∈ ˜l2
ψ,I
= dom Kλ,I. Let ˜v ∈ ˜l2
ψ,1
be determined by the
sequence v ∈ C([0, ∞)Z)2n with the terms
vk =
{
zk, k ∈ IZ,
0, k IZ.
Then Kλ,I(˜z) =
∑
k∈I Θ∗
k
(¯λ)ψk zk =
∑∞
k=0 Θ∗
k
(¯λ)ψk vk = Kλ(˜v). Hence Ran Kλ,I ⊆ Ran Kλ,
which upon combining with the first part of the proof yields Ran Kλ = Ran Kλ,I. Therefore
Ran Kλ = Ran ϑ(IZ) by Lemma 6.4.6. ■
– 111 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
Lemma 6.4.8. Let IZ = [0, ∞)Z and IZ ⊂ [0, ∞)Z be a finite discrete interval for which rank ϑ(IZ)
is maximal. Then
˜l2
ψ,1 = Ker Kλ ⊕
{
˜z ∈ ˜l2
ψ,1 | z = Θ(¯λ)ξ, ξ ∈ Ran ϑ(IZ)
}
(6.72)
and codim(Ker Kλ) = rank ϑ(IZ).
Proof. In the complete analogy with the arguments given in the proof of Lemma 6.4.6,
one can show that
˜l2
ψ,1 = Ker Kλ ⊕ (Ker Kλ)⊥
= Ker Kλ ⊕ Ran K∗
λ,
where
Ran K∗
λ =
{
˜z ∈ ˜l2
ψ,1 | z = Θ(¯λ)ξ, ξ ∈ Ran ϑ(IZ)
}
,
and that
(Ran Kλ)⊥
= Ker K∗
λ =
{
ξ ∈ C2n
| ||Θ(¯λ)ξ||ψ = 0
}
=
{
ξ ∈ C2n
| ψk Θk(¯λ)ξ = 0 for all k ∈ [0, ∞)Z
}
.
Finally, let ξ1, . . . , ξm denote a basis for Ran ϑ(IZ) = Ran Kλ and put v
[j]
k
:= Θk(¯λ)ξj for
all k ∈ [0, ∞)Z. Then ˜v[1]
, . . . , ˜v[m]
∈ Ran K∗
λ
and suppose that
∑m
j=1 cj ˜v[j]
= ˜0 ∈ ˜l2
ψ,1
for some
numbers c1, . . . , cm ∈ C. Then ψk Θk(¯λ)(
∑m
j=1 cj ξj) = 0 for all k ∈ [0, ∞)Z, which implies
m∑
j=1
cj ξj ∈ Ran Kλ ∩ (Ran Kλ)⊥
= {0}
by the first part. Hence c1 = · · · = cm = 0, i.e., ˜v[1]
, . . . , ˜v[m]
∈ Ran K∗
λ
are linearly independent
in ˜l2
ψ,1
. Thus codim(Ker Kλ) = dim(Ker Kλ)⊥ = dim(Ran K∗
λ
) = rank ϑ(IZ). ■
6.4.3 Minimal linear relation and its deficiency indices
Before we define the minimal linear relation Tmin and establish the fundamental relation
between Tmax and Tmin, we prove two auxiliary lemmas.
Lemma 6.4.9. We have Ker Kλ ⊆ Ran(T0 − λI) for every λ ∈ C.
Proof. Let λ ∈ C and ˜f ∈ Ker Kλ, i.e., Kλ( ˜f) =
∑
k∈I Θ∗
k
(¯λ)ψk zk = 0. Assume that IZ is
a finite discrete interval, i.e., IZ = [0, N]Z for some N ∈ N ∪ {0}. Then ˜f ∈ ˜l2
ψ
by (6.69) and
for any g ∈ ˜f the sequence z ∈ l2
ψ,0
with the terms
zk :=



0, k = 0,
Θk(λ)J
∑k−1
j=0 Θ∗
j
(¯λ)ψj gj, k ∈ I+
Z K{0},
(6.73)
solves the nonhomogeneous system L (z) = λψz + ψg on IZ, see also [61, Theorem 3.17].
Thus, ˜f ∈ Ran(T0 − λI) and the conclusion follows; cf. (6.64).
On the other hand, if IZ = [0, ∞)Z, then ˜f ∈ ˜l2
ψ,1
by (6.69), i.e., there exists K ∈ [0, ∞)Z
such that ψk gk = 0 for all k ∈ [K, ∞)Z and all g ∈ ˜f. Hence for any g ∈ ˜f the sequence
z ∈ l2
ψ
with the terms
zk := −Θk(λ)J
∞∑
j=k
Θ∗
j(¯λ)ψj gj =



−Θk(λ)J
∑K−1
j=k Θ∗
j
(¯λ)ψj gj, k ∈ [0, K − 1]Z,
0, k ∈ [K, ∞)Z,
(6.74)
– 112 –
6.4. Maximal and minimal linear relations
satisfies L (z) = λψz + ψg on [0, ∞)Z. In addition, z0 = −Θ0(λ)JKλ( ˜f) = 0, which implies
z ∈ l2
ψ,0
. Thus again ˜f ∈ Ran(T0 − λI) and the proof is complete. ■
Lemma 6.4.10. We have ⟨ ˜f, ˜y⟩ψ = ⟨˜z, ˜g⟩ψ for every {˜z, ˜f} ∈ T0 and {˜v, ˜g} ∈ Tmax.
Proof. Let {˜z, ˜f} ∈ T0 and {˜v, ˜g} ∈ Tmax, i.e., there are z ∈ l2
ψ,0
and v ∈ l2
ψ
such that
L (z) = ψ f and L (v) = ψg on IZ. Then, similarly as in (6.14), we get
k∑
j=0
z∗
j ψj gj = −(z, v)j
k+1
0
+
k∑
j=0
f∗
j ψj vj.
However, z ∈ l2
ψ,0
implies z0 = 0 and the existence of K ∈ I+
Z such that zk = 0 for all
k ∈ [K, ∞)Z ∩I+
Z . As a consequence we see that ⟨z, g⟩ψ = ⟨ f, v⟩ψ, and thus ⟨z, g⟩ψ = ⟨ f, v⟩ψ
for any z ∈ ˜z and v ∈ ˜v. ■
By Lemma 6.4.10 and the definition of the adjoint linear relation, see (A.4), one obtains
T0 ⊆ Tmax ⊆ T∗
0, (6.75)
from which we conclude that T0 is symmetric in ˜l2×2
ψ
, and hence closable. We then define
the minimal linear relation Tmin by
Tmin := T0. (6.76)
Another approach to the definition of a minimal linear relation is to put Tmin := T∗
max;
cf. [116, Definition 2.3] and [13, Identity (4.2)]. In the next theorem we establish the
equivalence of these approaches for the linear relations at hand. We note also that an
alternative demonstration of the next statement can be patterned after that presented
in [13, pp. 1354–1355] by using the results in Section 6.3 and [13, Proposition A.2].
Theorem 6.4.11. The linear relations T0, Tmax, and Tmin as defined in (6.63), (6.62), and (6.76),
respectively, satisfy
T∗
0 = T∗
min = Tmax. (6.77)
Proof. By (6.75), note that Tmax ⊆ T∗
0
= T0
∗
= T∗
min
. Thus it remains to show that T∗
0
⊆ Tmax;
or equivalently, given {˜z, ˜f} ∈ T∗
0
, that there is a z ∈ ˜z ∈ ˜l2
ψ
such that L (z) = ψf.
Let {˜z, ˜f} ∈ T∗
0
be given and r ∈ C(I+
Z )2n satisfy L (r) = ψf. Whenever {˜v, ˜g} ∈ T0, i.e.,
there exists v ∈ ˜v such that v ∈ l2
ψ,0
and L (v) = ψ g, we obtain from (6.14) that ⟨r, g⟩ψ =
⟨f, v⟩ψ, because v0 = 0 and vk = 0 for all sufficiently large k ∈ I+
Z . Simultaneously,
from the definition of T∗
0
, we have ⟨˜z, ˜g⟩ψ = ⟨ ˜f, ˜v⟩ψ. Therefore ⟨z − r, g⟩ψ = 0 for any
z ∈ ˜z ∈ dom T∗
0
and g ∈ ˜g ∈ Ran T0. Consequently, for any h ∈ ˜h ∈ Ker K ⊆ Ran T0, see
Lemma 6.4.9 with λ = 0, we get from the previous part that ⟨z − r, h⟩ψ = 0. Thus, for any
ξ ∈ C2n we have ∑
k∈I
(zk − rk − Θk ξ)∗
ψk hk = 0, (6.78)
because K(˜h) =
∑
k∈I Θ∗
k
ψk hk = 0 by (6.69).
Let IZ = [0, N]Z with N ∈ N ∪ {0}, then obviously ˜z, ˜r ∈ ˜l2
ψ
. Hence by (6.71) there exists
η ∈ Ran ϑ(IZ) such that the equivalence class corresponding to z−r−Θη belongs to Ker K.
But in that case it is equal to ˜0 by (6.78). Therefore ||z − r − Θη||ψ = 0, i.e., r + Θη ∈ ˜z, and
L (r + Θη) = L (r) = ψ f, which implies T∗
0
⊆ Tmax.
– 113 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
In the opposite case, i.e., IZ = [0, ∞)Z, let IZ ⊂ [0, ∞)Z be a finite discrete interval
such that k0 ∈ I
+
Z and the value of rank ϑ(IZ) = m ≤ 2n is maximal as noted in
Lemma 6.2.10. Then ˜l2
ψ,1
= Ker K⊕ Ker K⊥ and by Lemma 6.4.8 we see that there exists
a basis ˜v[1]
, . . . , ˜v[m]
∈ ˜l2
ψ,1
of the space Ker K⊥, in which v
[j]
k
:= Θk ξ[j]
on [0, ∞)Z for
j ∈ {1, . . . , m} with ξ[1]
, . . . , ξ[m]
forming a basis of Ran ϑ(IZ). Moreover, it follows from the
definition of ˜l2
ψ,1
that there is a finite discrete interval IZ = [0, K]Z such that IZ ⊆ IZ and
ψk v
[j]
k
= 0 for all k ∈ [K, ∞)Z and j ∈ {1, . . . , m}. (6.79)
If ˜d ∈ Ker KI := Ker K0,I, then for all j ∈ {1, . . . , m} we get
⟨v[j]
, d⟩ψ,I :=
∑
k∈I
ξ[j]∗
Θ∗
k ψk dk = ξ[j]∗
KI( ˜d) = 0,
which implies ˜v[1]
, . . . , ˜v[m]
∈ Ker K⊥
I
. Now, let s ∈ l2
ψ,1
be defined by
sk :=



zk − rk, k ∈ IZ,
0, k ∈ [0, ∞)Z KIZ,
where {˜z, ˜f} and r are the same as in the first part. Obviously ˜s ∈ ˜l2
ψ
and, by Lemma 6.4.6,
there is p ∈ Ker K⊥
I
, where p := Θη for some η ∈ Ran ϑ(IZ), such that ˜s − p ∈ Ker KI. As
a consequence, we obtain ⟨˜s − p, ˜v[j]
⟩ψ,I = 0 for all j ∈ {1, . . . , m}. Then, by (6.79), we have
for all j ∈ {1, . . . , m} that
∑
k∈I
(zk − rk − Θk η)∗
ψk v
[j]
k
=
∑
k∈I
(zk − rk − Θk η)∗
ψk v
[j]
k
= ⟨˜s − p, ˜v[j]
⟩ψ,I = 0. (6.80)
Hence the direct sum decomposition of ˜l2
ψ,1
in (6.72) and identities (6.78) and (6.80) show
that ⟨z − r − Θη, q⟩ψ = 0 for any q ∈ l2
ψ,1
; in particular, for
qk :=



zk1
− rk1
− Θk1
η, k = k1,
0, k ∈ [0, ∞)Z K{k1}.
Since k1 ∈ [0, ∞)Z can be chosen arbitrarily, it follows ψk (zk − rk − Θk η) ≡ 0 on [0, ∞)Z, i.e.,
||z − (r + Θη)||ψ = 0. Therefore, r + Θη ∈ ˜z and it satisfies L (r + Θη) = L (r) = ψf, which
again implies T∗
0
⊆ Tmax. ■
Moreover, the following theorem provides an explicit characterization of the minimal
linear relation Tmin; cf. [135, Theorem 3.2].
Theorem 6.4.12. Let system (Sλ) be definite on a finite discrete interval ID
Z := [a, b]Z ⊆ IZ, where
a, b ∈ IZ. Then
Tmin =
{
{˜z, ˜f} ∈ Tmax | ˆz0 = 0 = (ˆz, ˆv)N+1 for all ˜v ∈ dom Tmax
}
, (6.81)
which in the case of IZ being a finite discrete interval reduces to
Tmin =
{
{˜z, ˜f} ∈ Tmax | ˆz0 = 0 = ˆzN+1
}
. (6.82)
– 114 –
6.4. Maximal and minimal linear relations
Proof. Since Tmin = (Tmin) by the definition, identities (A.11), (6.67), and (6.77) yield
Tmin =
{
{˜z, ˜f} ∈ Tmax | (ˆz, ˆv)k
N+1
0
= 0 for all ˆv ∈ dom Tmax
}
. (6.83)
Let T be the linear relation on the right-hand side of (6.81). Then obviously T ⊆ Tmin. On
the other hand, let {˜z, ˜f} ∈ Tmin be fixed. Then, (ˆz, ˆv)k
N+1
0
= 0 for all ˆv ∈ dom Tmax by (6.83).
By Lemma 6.4.3, for any {˜v, ˜g} ∈ Tmax there exists {˜r, ˜h} ∈ Tmax such that ˆvk = 0 for k ∈ [0, c]Z
and ˆrk = ˆvk for k ∈ [d + 1, ∞)Z ∩ I+
Z . Hence (ˆz, ˆv)0 = (ˆz, ˆv)N+1 = 0 for all ˆv ∈ dom Tmax. From
the second part of Lemma 6.4.3 we get ˆz0 = 0, because there exists {˜z[i]
, ˜f[i]
} ∈ Tmax such that
ˆz[i]
0
= ei. Therefore T = Tmin. If, in addition, IZ is a finite discrete interval, i.e., IZ = [0, N]Z
with N ∈ N ∪ {0}, then dom Tmax contains also ˜r such that ˆrN+1 = ei, i ∈ {1, . . . , 2n}, by the
last part of Lemma 6.4.3. Hence equality (6.82) holds. ■
Finally, we define the defect subspace and defect index for the minimal linear relation
Tmin according to (A.7) and (A.8), respectively, i.e., by using (6.77) we put
Nλ = Nλ(Tmin) :=
{
˜z ∈ ˜l2
ψ | {˜z, λ˜z} ∈ T∗
min = Tmax
}
, ˜nλ := dim Nλ.
Since Tmin is a closed, symmetric linear relation by Theorem 6.4.11, it follows that the
value of the deficiency indices ˜nλ is constant on each of the open upper and lower halfplanes
of C, i.e., ˜n±λ = ˜n±i for all λ ∈ C+; cf. [143, Theorem 2.13]. Thus we let ˜n± := ˜n±i.
In addition, we introduce the following two subspaces
Nλ :=
{
z ∈ l2
ψ | L (z)k = λψk zk for all k ∈ IZ
}
, nλ := dim Nλ,
kλ = kλ(Tmin) :=
{
{˜z, λ˜z} | {˜z, λ˜z} ∈ Tmax
}
.
In other words, the subspace Nλ ⊆ l2
ψ
consists of all square summable solutions of system
(Sλ) and nλ denotes the number of linearly independent square summable solutions
of system (Sλ), while kλ is a subspace associated with the defect subspace Nλ. Then we
obtain the following von Neumann decomposition of the maximal linear relation Tmax in
terms of the minimal linear relation Tmin and the subspaces kλ and k¯λ, i.e.,
Tmax = Tmin ∔ kλ ∔ k¯λ, (6.84)
where the direct sum ∔ is orthogonal if λ = ±i, see (A.9).
Assuming that rank ϑ(IZ) is maximal, as described in Lemma 6.2.10, we next show
a relationship between the numbers nλ and ˜nλ.
Theorem 6.4.13. Let IZ ⊆ IZ be a finite discrete interval such that the value of rank ϑ(IZ) is
maximal. Then, for any λ ∈ C,
nλ = ˜nλ + 2n − rank ϑ(IZ). (6.85)
Proof. Let λ ∈ C be given and π1 denote the restriction of the quotient space map π
(introduced at the beginning of this section) given by π1 := π|Nλ
: Nλ → Nλ. Since for any
˜z ∈ Nλ there exists z ∈ ˜z solving system (Sλ), i.e., z ∈ Nλ, the map π1 is surjective. Thus
dim Nλ = dim Nλ + dim(Ker π1). (6.86)
Moreover, we note that
Ker π1 =
{
z ∈ l2
ψ | L (z)k = λψk zk and ψk zk = 0 for all k ∈ IZ
}
,
– 115 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
which shows that z ∈ Ker π1 if and only if z = Θξ for some ξ ∈
∩N
K=0 Ker ϑ([0, K]Z) =
Ker ϑ(IZ). Therefore dim(Ker π1) = dim[Ker ϑ(IZ)] = 2n − rank ϑ(IZ), which together
with equality (6.86) implies (6.85). ■
As a direct consequence of equality (6.85) we get the following properties of the
numbers nλ and ˜nλ.
Corollary 6.4.14. The following statements hold true:
(i) nλ − ˜nλ is nonnegative and constant for all λ ∈ C;
(ii) nλ = ˜nλ for some (and hence for any) λ ∈ C if and only if system (Sλ) is definite on [0, ∞)Z;
(iii) nλ is constant in the half-planes C+ and C−.
Proof. The first statement follows from equality (6.85) and Lemma 6.2.9, while for the
second statement it suffices to combine (6.85), Theorem 6.2.11, and Lemma 6.2.3. Finally,
since the minimal linear relation Tmin is symmetric, see (6.75) and (6.76), the number ˜nλ
is constant in the half-planes C+ and C− by [143, Theorem 2.13]. Hence the value of nλ is
also constant in C+ and C− by the first part. ■
Remark 6.4.15. The last statement of Corollary 6.4.14 extends the enumeration and analysis
of linearly independent square summable solutions of system (Sλ) provided in Section
2.4, see Theorem 2.4.8 and also equality (6.3). More precisely, since the number of
linearly independent square summable solutions of system (Sλ) or (Sλ) is constant in C+
and C−, the the same property possesses also the function r(λ) defined in (2.56) as it
was already mentioned in Remark 2.4.21. Moreover, in the final result of this chapter
we complete this analysis by a basic estimate, which shows that the number of linearly
independent square summable solutions on the real line cannot exceed the same number
calculated for any λ ∈ CKR.
Theorem 6.4.16. For any λ ∈ C we have
Ker(Tmin − λI) = {˜0}. (6.87)
Moreover, for every λ ∈ R we have ˜nλ ≤ ˜n± and nλ ≤ n±i.
Proof. First, suppose that IZ = [0, N]Z for some N ∈ N ∪ {0}. Let λ ∈ C, ˜f ∈ ˜l2
ψ
, and
˜z ∈ ˜l2
ψ
be determined by z ∈ C(I+
Z )2n constructed as in (6.73). Then L (z) = λψz + ψf on
IZ, which yields {˜z, ˜f} ∈ Tmax − λI, i.e., ˜f ∈ Ran(Tmax − λI). Hence ˜l2
ψ
= Ran(Tmax − λI),
because λ and ˜f were chosen arbitrarily. Thus
Ker(Tmin − λI)
(A.5)
= [Ran(T∗
min − ¯λI)]⊥ (6.77)
= [Ran(Tmax − ¯λI)]⊥
= ( ˜l2
ψ)⊥
= {˜0}.
On the other hand, let now IZ = [0, ∞)Z. Then for λ ∈ C, ˜f ∈ ˜l2
ψ,1
, and ˜z ∈ ˜l2
ψ,1
determined by the sequence z ∈ C([0, ∞)Z)2n constructed as in (6.74) we obtain that
{˜z, ˜f} ∈ Tmax − λI, see the proof of Lemma 6.4.9. Hence ˜l2
ψ,1
⊆ Ran(Tmax − λI). Since ˜l2
ψ,1
is a dense subspace of ˜l2
ψ
, it follows that Ran(Tmax − λI) is also dense in ˜l2
ψ
for any λ ∈ C.
Thus, for the proof of equality (6.87) it suffices to utilize an analogous calculation as in the
previous part. The remaining assertions follow from (A.10) and Corollary 6.4.14(i). ■
– 116 –
6.5. Bibliographical notes
6.5 Bibliographical notes
The results of this chapter were published in [A18] except for Remark 6.1.4, Lemmas 6.2.8
and 6.4.3, and Theorem 6.4.12, which were published later in [A21]. Their generalization
to symplectic systems on time scales is one of the goals of the current research.
– 117 –
Chapter 6. Nohomogeneous problem and maximal and minimal linear relations
– 118 –
Chapter 7
Self-adjoint extensions
... when America’s National Academy of Science asked shortly before
his death what he thought were his three greatest achievements ...
Johnny replied to the academy that he considered his most important
contributions to have been on the theory of self-adjoint operators in
Hilbert space, and on the mathematical foundations of quantum theory
and the ergodic theorem.
John von Neumann, see [118, pg. 24]
This final chapter is devoted to the characterization of all self-adjoint extensions of the
minimal linear relation Tmin defined in (6.76). The description of self-adjoint extensions
and their particular cases is a classic problem in the theory of differential and difference
equations, see [14,27,44,51,60,80,81,95,98,119,124,126,127,147,159,160,162,169] and [A5].
As in [135,160,162,169], our main result is obtained by using square summable solutions
of system (Sλ) and the Glazman–Krein–Naimark theory, see again the Appendix of this
thesis for general results from the latter theory.
Throughout this chapter we consider system (Sλ) with the coefficients specified in
Notation 6.1.2 and it is organized as follows. In Section 7.1 we establish a limit point
criterion for system (Sλ), see Theorem 7.1.1. In Section 7.2 we present the main result,
Theorem 7.2.1, concerning the characterization of self-adjoint extensions of the minimal
linear relation associated with system (Sλ). We apply this to a consideration of the 2 × 2
(scalar) case for a finite discrete interval and describe the Krein–von Neumann extension
explicitly, see Theorems 7.2.7 and 7.2.9, and Example 7.2.8. We note that there is no
analogue of Theorems 7.1.1, 7.2.7, 7.2.9 and Example 7.2.8 in the setting of system (2.6).
Finally, Section 7.3 is devoted to the proof of Theorem 7.2.1.
7.1 Limit point criterion
As noted in the Appendix, the equality ˜n+ = ˜n− is necessary and sufficient for the
existence of a self-adjoint extension of the minimal linear relation Tmin. Since the latter
equality is equivalent with n+ = n− by Corollary 6.4.14(i), we need to guarantee that the
number of linearly independent square summable solutions of system (Sλ) is constant on
the set CKR. This condition is trivially satisfied if the discrete interval IZ is finite or if
system (Sλ) is in the limit circle case for some (and hence for all) λ ∈ C, i.e., nλ = 2n, while it
can be violated in any other case nλ < 2n with IZ = [0, ∞)Z, see Remarks 2.4.21 and 6.4.15.
Moreover, it was also discussed in Remark 2.4.16 that the classical limit point criterion for
linear Hamiltonian differential and difference systems (2.5) and (2.6) utilizes the minimal
eigenvalue of the corresponding weight matrix and a similar criterion cannot be derived
– 119 –
Chapter 7. Self-adjoint extensions
in the current setting. Nevertheless, in the following theorem we give conditions, which
imply the invariance of the limit point case on CKR for system (Sλ) with the special linear
dependence on λ specified in (6.25), i.e., for system (6.26); cf. [134, Theorem 6.2]. This
statement is a discrete analogue of [116, Theorem 5.6].
Theorem 7.1.1. Let IZ = [0, ∞)Z and consider system (Sλ) with the coefficients given in (6.25),
where B∗
k
Ck ≡ 0, B∗
k
Dk > 0, and Wk > 0 for all k ∈ IZ. If there exists h ∈ C(IZ)1 such that
hk ≥ h > 0 and
A∗
k Ck ≥ −hk Wk+1,
∞∑
k=0
1
gk
√
hk
= ∞, (7.1)
where gk := max
{
1, W−1/2
k+1
(B∗
k
Dk)−1/2
2
}
, and a constant T ≥ 0 such that
(
1
hk
)
gk ≤
T
√
hk
for all k ∈ IZ, (7.2)
then system (Sλ) is in the limit point case for all λ ∈ CKR, i.e., nλ = n for all λ ∈ CKR.
Proof. System (Sλ) with the coefficients from (6.25) can be written as (6.26) and the invertibility
of Bk and Wk on IZ implies the definiteness of this system by Theorem 6.2.5.
In accordance with Theorem 2.4.3, we have nλ = n if and only if Zk(λ)ξ l2
ψ
for any
ξ ∈ CnK{0}, where Z(λ) is the same as in Section 6.3, i.e., the 2n × n solution of system
(Sλ) determined by the initial condition Z0(λ) = −Jα∗ with a given α ∈ . Then
zk :=
( xk
uk
)
= Zk(λ)ξ with x, u ∈ C([0, ∞)Z)n satisfies z∗
0
Jz0 = 0. Moreover, it is sufficient
to consider only λ = ±i, because the number nλ ≥ n is constant in C+ and C− by Corollary
6.4.14(iii). Hence, let ξ ∈ CnK{0} and λ ∈ {±i} be fixed. We show that under the
current assumptions we have z l2
ψ
.
Let us assume that z ∈ l2
ψ
. By a direct calculation, we obtain from the block structure
of the system at hand, see (6.26), and from the symplecticity of the matrix Sk, see
identities (1.17) and (1.18), that
(x∗
k uk) = −x∗
k+1A∗
k Ck xk+1 − x∗
k+1 C∗
k Bk uk+1 − u∗
k+1B∗
k Ck xk+1 − u∗
k+1B∗
k Dk uk+1 − λx∗
k Wk xk.
Since B∗
k
Dk > 0 and hk > 0, the quantity Fk(x, u) :=
( ∑k
j=0
1
hj
u∗
j+1
B∗
j
Dj uj+1
)1/2
≥ 0 is
well-defined. Then the latter equality and the assumption B∗
k
Ck ≡ 0 yield
F2
k (x, u) = −
k∑
j=0
1
hj
x∗
j+1 A∗
j Cj xj+1 − λ
k∑
j=0
1
hj
x∗
j Wj xj −
k∑
j=0
1
hj
(x∗
j uj). (7.3)
From the Hermitian property and positive definiteness of Wk and B∗
k
Dk, the Cauchy–
Schwarz inequality, inequality (1.8), and the definition of gk we obtain
| x∗
k+1 uk+1 | =
(
W1/2
k+1
xk+1
)∗
W−1/2
k+1
(B∗
k Dk)−1/2
(B∗
k Dk)1/2
uk+1
≤ W1/2
k+1
xk+1 2
× W−1/2
k+1
(B∗
k Dk)−1/2
(B∗
k Dk)1/2
uk+1 2
≤ W1/2
k+1
xk+1 2
× W−1/2
k+1
(B∗
k Dk)−1/2
σ
× (B∗
k Dk)1/2
uk+1 2
≤ gk W1/2
k+1
xk+1 2
× (B∗
k Dk)1/2
uk+1 2
. (7.4)
– 120 –
7.1. Limit point criterion
Hence the latter inequality, assumption (7.2), the Cauchy–Schwarz inequality, and the
inequality of arithmetic and geometric means
√
ab ≤ a+b
2 yield
k∑
j=0
(
1
hj
)
x∗
j+1 uj+1 ≤
k∑
j=0
(
1
hj
)
| x∗
j+1 uj+1 | ≤
k∑
j=0
(
1
hj
)
gj W1/2
j+1
xj+1 2
× (B∗
j Dj)1/2
uj+1 2
≤
k∑
j=0
T W1/2
j+1
xj+1 2
× h−1/2
j
(B∗
j Dj)1/2
uj+1 2
≤
(
T2
k∑
j=0
W1/2
j+1
xj+1
2
2
)1/2
×
( k∑
j=0
h−1
j (B∗
j Dj)1/2
uj+1
2
2
)1/2
≤
1
2
(
T2
k∑
j=0
W1/2
j+1
xj+1
2
2
+
k∑
j=0
h−1
j (B∗
j Dj)1/2
uj+1
2
2
)
≤ 1
2
(
T2
||z||2
ψ + F2
k (x, u)
)
. (7.5)
By using the summation by parts together with the assumption hk ≥ h and the inequalities
from (7.4) and (7.5) we get
re
[ k∑
j=0
1
hj
(x∗
j uj)
]
≤
k∑
j=0
1
hj
(x∗
j uj) ≤
[
x∗
j uj/hj
]k+1
0
−
k∑
j=0
(
1
hj
)
x∗
j+1 uj+1
≤ | x∗
0 u0/h0 | + | x∗
k+1 uk+1/hk+1 | +
k∑
j=0
(
1
hj
)
x∗
j+1 uj+1
≤ T1 +
1
h
gk W1/2
k+1
xk+1 2
× (B∗
k Dk)1/2
uk+1 2
+ 1
2
(
T2
||z||2
ψ + F2
k (x, u)
)
, (7.6)
where T1 := | x∗
0
u0/h0 |. Since
re
(
F2
k (x, u)
)
= −
k∑
j=0
1
hj
x∗
j+1 A∗
j Cj xj+1 − re
( k∑
j=0
1
hj
(x∗
j uj)
)
and the inequality in (7.1) implies
−
k∑
j=0
1
hj
x∗
j+1 A∗
j Cj xj+1 ≤
k∑
j=0
x∗
j+1 Wj xj+1 ≤ ||z||2
ψ,
it follows from (7.3) and (7.6) that
1
2
k∑
j=0
g−1
j h−1/2
j
F2
j (x, u) ≤ T2
k∑
j=0
g−1
j h−1/2
j
+
1
h
k∑
j=0
h−1/2
j
W1/2
j+1
xj+1 2
× (B∗
j Dj)1/2
uj+1 2
,
where we put T2 := T1 + (1 + T2/2)||z||2
ψ. Then with the aid of the Cauchy–Schwarz
inequality we obtain
Gk :=
1
2
k∑
j=0
g−1
j h−1/2
j
[F2
j (x, u) − 2T2] ≤
1
h
( k∑
j=0
W1/2
j+1
xj+1
2
2
)1/2
×
( k∑
j=0
(B∗
j Dj)1/2
uj+1
2
2
)1/2
≤
1
h
||z||ψ Fk(x, u). (7.7)
– 121 –
Chapter 7. Self-adjoint extensions
In the next part we show that F2
k
(x, u) ≤ 2T2 for all k ∈ IZ. Assume that there exists
an index m ∈ IZ such that F2
m(x, u) > 2T2. Since F2
k
(x, u) is nondecreasing, we have
F2
k
(x, u) − 2T2 > t for all k ∈ [m, ∞)Z, where t := F2
m(x, u) − 2T2. Also Gk is nondecreasing
for all k ∈ [m − 1, ∞)Z and for all k ∈ [m, ∞)Z we get from inequality (7.7) and the relation
F2
k
(x, u) = 2 gk h1/2
k
Gk−1 + 2T2 that
h2
− 2||z||2
ψ G−2
k T2 ≤ 2G−2
k ||z||2
ψ gk h1/2
k
Gk−1. (7.8)
In addition, Gk ≥ t
2
∑k
j=0 g−1
j
h−1/2
j
→ ∞ as k → ∞ by the second part of (7.1). Now, let
0 < a < 2h2 be arbitrary and ℓ ∈ [m, ∞)Z be such that Gℓ ≥ 2||z||ψ T1/2
2
/
√
2h2 − a. Then we
have a/2 ≤ h2 − 2G−2
k
T2 ||z||2
ψ for all k ∈ [ℓ, ∞)Z, which together with (7.8) yields for any
k ∈ [ℓ + 1, ∞)Z that
a
2
k∑
j=ℓ+1
1
gj h1/2
j
≤
k∑
j=ℓ+1
1
gj h1/2
j
(
h2
− 2G−2
j T2 ||z||2
ψ
)
≤
k∑
j=ℓ+1
2G−2
j ||z||2
ψ Gj−1
≤ 2||z||2
ψ
k∑
j=ℓ+1
Gj−1
Gj Gj−1
≤ −2||z||2
ψ
k∑
j=ℓ+1
(
1
Gj−1
)
≤ 2||z||2
ψ
1
Gℓ
< ∞.
But it contradicts the second assumption from (7.1) as k → ∞. Thus it holds F2
k
(x, u) ≤ 2T2
for all k ∈ IZ, i.e.,
∞∑
j=0
h−1
j u∗
j+1 B∗
j Dj uj+1 ≤ 2T2 < ∞. (7.9)
Since system (Sλ) is definite on [0, ∞)Z, there exists s ∈ IZ such that
∑s
j=0 z∗
k
ψk zk = T3 > 0.
Hence the positive definiteness of Wk and the extended Lagrange identity in (6.11) yield
z∗
k+1 Jzk+1 = z∗
0 Jz0 ± 2i
k∑
j=0
z∗
j ψj zj = 2
k∑
j=0
z∗
j ψj zj ≥ 2
s∑
j=0
z∗
j ψj zj = 2T3, (7.10)
for any k ∈ [s, ∞)Z. Simultaneously, we get from (7.4) the estimate
z∗
k+1 Jzk+1 ≤ 2 x∗
k+1 uk+1 ≤ 2 gk h1/2
k
|| W1/2
k+1
xk+1 ||2 × h−1/2
j
||(B∗
k Dk)1/2
uk+1 ||2. (7.11)
Inequalities (7.9), (7.10), (7.11) together with the Cauchy–Schwarz inequality imply for
any k ∈ [s, ∞)Z that
k∑
j=s
1
gj h1/2
j
≤
k∑
j=s
2
z∗
j+1
Jzj+1
W1/2
k+1
xk+1 2
× h−1/2
j
(B∗
k Dk)1/2
uk+1 2
≤
1
T3
k∑
j=s
W1/2
k+1
xk+1 2
× h−1/2
j
(B∗
k Dk)1/2
uk+1 2
≤
1
T3
( k∑
j=s
W1/2
k+1
xk+1
2
2
)1/2
×
( k∑
j=s
h−1
j (B∗
k Dk)1/2
uk+1
2
2
)1/2
≤
1
T3
||z||ψ
√
2 T1/2
2
< ∞,
– 122 –
7.2. Main results
which (again) contradicts the second condition in (7.1) as k → ∞. Hence z l2
ψ
. Since ξ
and λ were chosen arbitrarily, it follows that Z(λ)ξ l2
ψ
for any ξ ∈ CnK{0}. Therefore,
system (Sλ) is in the limit point case for λ ∈ {±i} and consequently for all λ ∈ CKR. ■
Upon applying Theorem 7.1.1 to system (6.22) with qk ≡ 0 we obtain the following
corollary for a special case of the second order Sturm–Liouville difference equation (6.24),
because one easily observes that z(λ) ∈ l2
ψ
if and only if
∑∞
k=0 |yk(λ)|2 wk < ∞, where zk(λ)
are ψk are as in Example 6.2.4.
Corollary 7.1.2. Let IZ = [0, ∞)Z and consider equation (6.24) with qk ≡ 0, pk < 0 and wk > 0
for all k ∈ IZ. If there exist hk ∈ C(IZ)1 and a constant T ≥ 0 such that hk ≥ h > 0 and
∞∑
k=0
1
gk
√
hk
= ∞,
(
1
hk
)
gk ≤
T
√
hk
, k ∈ IZ,
where gk := max
{
1,
(
−
pk+1
wk+1
)1/2}
, then equation (6.24) is in the limit point case for any λ ∈ CKR,
i.e., there exists only one nontrivial solution satisfying
∑∞
k=0 |yk(λ)|2 wk < ∞.
It was shown in [96, Theorem 10], see also [157, Corollary 3.1], that equation (6.24)
with pk 0 and wk > 0 is in the limit point case for any λ ∈ CKR if
∑∞
k=0
(wk wk+1)1/2
| pk+1 | = ∞.
Corollary 7.1.2 partially generalizes this classical limit-point criterion as shown in the
following example; cf. [A23, Example 3.4].
Example 7.1.3. Let us consider the equation
(6.24), pk ≡ −1, qk ≡ 0, wk = 1/(k + 1)2
, IZ = [0, ∞)Z. (7.12)
Then the criterion from [96, Theorem 10] cannot be applied, because
∞∑
k=0
(
wk wk+1
)1/2
| pk+1 |
=
∞∑
k=0
√
1
(k + 1)2 (k + 2)2
= 1 < ∞.
On the other hand, the assumptions of Corollary 7.1.2 are satisfied with hk ≡ 1, gk = k + 2,
and T = 0, i.e., equation (7.12) is in the limit point case for all λ ∈ CKR. This fact
can be verified directly by using the Weyl alternative, see e.g. [9, Theorem 5.6.1] or
Corollary 2.4.23. More precisely, equation (7.12) with λ = 0 possesses two linearly
independent solutions y[1]
k
≡ 1 and y[2]
k
= k for k ∈ IZ ∪ {−1}. Since only y[1]
is square
summable with respect to wk, it follows from the Weyl alternative that equation (7.12) has
to be in the limit point case for all λ ∈ CKR. ▲
7.2 Main results
According to Corollary 6.4.14(iii), the number of linearly independent square summable
solutions of system (Sλ) is constant in the upper and lower half-planes of C. Hence the
numbers n+ := nλ for λ ∈ C+ and n− := nλ for λ ∈ C− are well-defined. Let λ0 ∈ C+ be
fixed. Then system (Sλ0
) has n+ linearly independent square summable solutions, which
we denote by s[1]
(λ0), . . . , s[n+]
(λ0), and similarly system (S¯λ0
) has n− linearly independent
square summable solutions, which we denote by s[1]
(¯λ0), . . . , s[n−]
(¯λ0). Let
φ[i]
k
:= s[i]
k
(λ0), φ
[j+n+]
k
:= s
[j]
k
(¯λ0), i ∈ {1, . . . , n+}, j ∈ {1, . . . , n−}, k ∈ I+
Z , (7.13)
Θ+
k := (φ[1]
k
, . . . , φ[n+]
k
), Θ−
k := (φ[1+n+]
k
, . . . , φ[p]
k
), and p := n+ + n−.
– 123 –
Chapter 7. Self-adjoint extensions
Note that 2n ≤ p ≤ 4n. Moreover, if system (Sλ) is definite on IZ, then the solutions
φ[1]
, . . . , φ[p]
belong to different equivalence classes. Hence for all i ∈ {1, . . . , n+} and
j ∈ {n+ + 1, . . . ,p} we have {φ[i]
, λ0 φ[i]
} ∈ Tmax and {φ[j]
, ¯λ0 φ[j]
} ∈ Tmax with φ[ℓ]
≡ φ[ℓ]
for
ℓ ∈ {1, . . . ,p}. We also define the matrix :=
(
(Θ+, Θ−), (Θ+, Θ−)
)
N+1
, i.e.,
=
( [1,1] [1,2]
[2,1] [2,2]
)
=


(φ[1]
, φ[1]
)N+1 . . . (φ[1]
, φ[p]
)N+1
...
...
...
(φ[p]
, φ[1]
)N+1 . . . (φ[p]
, φ[p]
)N+1


∈ Cp×p
, (7.14)
where [1,2]
∈ Cn+×n− . The elements ωij := (φ[i]
, φ[j]
)N+1 exist finite for all i, j ∈ {1, . . . ,p} by
identity (6.14). Furthermore, from (6.12) one easily concludes that the matrix [1,2]
consists
of the elements (φ[i]
, φ[j]
)N+1 = (φ[i]
, φ[j]
)0 for i ∈ {1, . . . , n+} and j ∈ {n+ + 1, . . . ,p}.
Identity (6.84) implies that for any {˜z, ˜f} ∈ Tmax we have
ˆzk = ˆvk +
p∑
j=1
ξj φ
[j]
k
, k ∈ I+
Z , (7.15)
where ˆv ∈ dom Tmin and ξ1, . . . , ξp ∈ C are determined uniquely. Especially, for the pairs
{˜z[1]
, ˜f[1]
}, . . . , {˜z[2n]
, ˜f[2n]
} ∈ Tmax (see Lemma 6.4.3), we get the unique expression
ˆz[i]
k
= ˆv[i]
k
+
p∑
j=1
ξi,j φ
[j]
k
, k ∈ I+
Z , i ∈ {1, . . . , 2n}. (7.16)
If we put Zk := (ˆz[1]
k
, . . . , ˆz[2n]
k
) for k ∈ I+
Z , then identity (7.16) implies
Zk = Vk + (Θ+
k , Θ−
k ) ⊤
, (7.17)
where Vk := (ˆv[1]
k
, . . . , ˆv[2n]
k
) ∈ C(I+
Z )2n×2n and the matrix ∈ C2n×p consists of ξi,j. In
particular, for k = 0 we obtain I = V0 + (Θ+
0
, Θ−
0
) ⊤, which together with (6.81) yields
I = (Θ+
0
, Θ−
0
) ⊤, i.e., rank = 2n by the second inequality in (1.3). From the definition
of ˆz[i]
(see Lemma 6.4.3), its expression in (7.16), and identity (6.81) we have
0 = (ˆz[i]
, φ[ℓ]
)N+1 = (ˆv[i]
, φ[ℓ]
)N+1 +
p∑
j=1
ξi,j (φ[j]
, φ[ℓ]
)N+1 =
p∑
j=1
ξi,j (φ[j]
, φ[ℓ]
)N+1
for all i ∈ {1, . . . , 2n} and any ℓ ∈ {1, . . . ,p}, i.e., = 0. Since rank = 2n, the first
inequality in (1.3) implies
rank ≤ p − 2n.
On the other hand, the equality [1,2]
= Θ+∗
0
JΘ−
0
and the first inequality in (1.3) yield
rank [1,2]
≥ p − 2n.
Therefore rank = p − 2n = rank [1,2]
. Since p − 2n ≤ n+ and p − 2n ≤ n−, we may
assume, without loss of generality, that φ[1]
, . . . , φ[n+]
are arranged such that
rank [1,2]
p−2n, n−
= p − 2n. (7.18)
The main result concerning the characterization of all self-adjoint extension of Tmin is
stated in the following theorem and its proof is given in Section 7.3; cf. [135, Theorem 5.7].
Recall that for the existence of a self-adjoint extension it is essential to have n+ = n−.
– 124 –
7.2. Main results
Theorem 7.2.1. Let system (Sλ) be definite on the discrete interval IZ, equality n+ = n− =: q
hold and assume that the solutions φ[1]
, . . . , φ[q]
are arranged such that (7.18) holds. Then a linear
relation T ⊆ ˜l2×2
ψ
is a self-adjoint extension of Tmin if and only if there exist matrices M ∈ Cq×2n
and L ∈ Cq×(2q−2n) such that
rank(M, L) = q, MJM∗
− L 2q−2n L∗
= 0, (7.19)
and
T =



{˜z, ˜f} ∈ Tmax | M ˆz0 − L


(φ[1], ˆz)N+1
...
(φ[2q−2n], ˆz)N+1

 = 0



. (7.20)
Remark 7.2.2. If, in addition to the assumptions of Theorem 7.2.1, there exists ν ∈ R such
that (Sν) has q linearly independent square summable solutions θ[1]
, . . . , θ[q]
(we suppress
the argument ν), then the statement of Theorem 7.2.1 can be formulated by using these
solutions, which are (without loss of generality) arranged such that the submatrix ϒ2q−2n
has the full rank, where
ϒ :=


(θ[1]
, θ[1]
)N+1 . . . (θ[1]
, θ[q]
)N+1
...
...
...
(θ[q]
, θ[1]
)N+1 . . . (θ[q]
, θ[q]
)N+1


,
see Lemma 7.3.3. Moreover, the Wronskian-type identity (6.12) yields that ϒ = ∗
0
J 0,
where k := (θ[1]
k
, . . . , θ
[q]
k
) for k ∈ I+
Z .
In the next part we discuss several special cases of Theorem 7.2.1. If system (Sλ) is in
the limit point case for all λ ∈ CKR, i.e., n+ = n− = n, then the boundary conditions at
N + 1 (which is necessary equal to ∞) are superfluous as stated in the following corollary;
cf. [135, Theorem 5.9]. This situation occurs, e.g., when the assumptions of Theorem 7.1.1
are satisfied. The proof follows directly from Theorem 7.2.1.
Corollary 7.2.3. Let system (Sλ) be definite on the discrete interval IZ and n+ = n− = n hold.
Then a linear relation T ⊆ ˜l2×2
ψ
is a self-adjoint extension of Tmin if and only if there exists a matrix
M ∈ Cn×2n such that
rank M = n, MJM∗
= 0,
and
T =
{
{˜z, ˜f} ∈ Tmax | M ˆz0 = 0
}
.
As it was already discussed in the previous chapters, if there exists λ0 ∈ C with
the property nλ0
= 2n, then system (Sλ) is in the limit circle case for all λ ∈ C, i.e.,
n+ = n− = 2n. Hence for any ν ∈ R there exist solutions θ[1]
, . . . , θ[2n]
(we again suppress
the argument ν) of system (Sν), which are linearly independent, square summable, and
the fundamental matrix ∈ C(I+
Z )2n×2n satisfies 0 = I, which implies ϒ = J, i.e.,
rank ϒ = 2n, see Remark 7.2.2. Upon combining the latter remark and Theorem 7.2.1 we
obtain the following result; cf. [135, Theorem 5.10].
Corollary 7.2.4. Let system (Sλ) be definite on the discrete interval IZ, ν ∈ R be fixed, and
assume that there exists a number λ0 ∈ C such that nλ0
= 2n. Let ∈ C(I+
Z )2n×2n be the
fundamental matrix of system (Sν) satisfying 0 = I and denote its columns by θ[1]
, . . . , θ[2n]
, i.e.,
k = (θ[1]
k
, . . . , θ[2n]
k
). Then a linear relation T ⊆ ˜l2×2
ψ
is a self-adjoint extension of Tmin if and
only if there exist matrices M, L ∈ C2n×2n such that
rank(M, L) = 2n, MJM∗
− LJL∗
= 0, (7.21)
– 125 –
Chapter 7. Self-adjoint extensions
and
T =



{˜z, ˜f} ∈ Tmax | M ˆz0 − L


(θ[1], ˆz)N+1
...
(θ[2n], ˆz)N+1

 = 0



. (7.22)
Especially, if IZ is a finite discrete interval, then the equality nλ = 2n is trivially
satisfied for any λ ∈ C. Therefore we get from Corollary 7.2.4 yet one more special case
of Theorem 7.2.1.
Corollary 7.2.5. Let IZ be a finite discrete interval and system (Sλ) be definite on IZ. Then
a linear relation T ⊆ ˜l2×2
ψ
is a self-adjoint extension of Tmin if and only if there exist matrices
M, L ∈ C2n×2n such that
rank(M, L) = 2n, MJM∗
− LJL∗
= 0, (7.23)
and
T = TM,L :=
{
{˜z, ˜f} ∈ Tmax | M ˆz0 − L ˆzN+1 = 0
}
. (7.24)
Proof. By Corollary 7.2.4 every self-adjoint extension of Tmin can be expressed as in (7.22)
with matrices M, L ∈ C2n×2n satisfying (7.21). If we put ˜L := L ∗
N+1
J ∈ C2n×2n, then the
matrices M, ˜L satisfies (7.23) and the linear relation in (7.22) can be written as TM, ˜L. ■
One can easily observe that a linear relation TM,L, i.e., the linear relation given by (7.24)
with M, L ∈ C2n×2n satisfying (7.23), is the same as a linear relation TM, L determined by
the matrices M := CM and L := CL for an arbitrary invertible matrix C ∈ C2n×2n. We show
that the converse is also true, see Remark 7.2.10(i). Moreover, it is well known that all selfadjoint
extensions of operators associated with the regular second order Sturm–Liouville
differential equations can be expressed by using the separated or coupled boundary
conditions, see e.g. [38]. In the last part of this section we show similar results for scalar
symplectic systems on a finite interval, i.e., n = 1 and IZ = [0, N]Z with N ∈ N ∪ {0},
and provide a unique representation of all self-adjoint extensions of Tmin. The main
assumptions for this treatment are summarized in the following hypothesis.
Hypothesis 7.2.6. The discrete interval IZ is finite, i.e., there exists N ∈ N ∪ {0} such that
IZ = [0, N]Z, we have n = 1, system (Sλ) is definite on IZ, and the matrices M, L ∈ C2×2 are
such that (7.23) holds.
If Hypothesis 7.2.6 is satisfied, then identity (7.23) implies that
either rank M = rank L = 2 or rank M = rank L = 1.
Hence we get the following dichotomy on the boundary conditions in (7.24).
Theorem 7.2.7. Let Hypothesis 7.2.6 be satisfied. Then the following statements hold.
(i) A linear relation TM,L given through M, L ∈ C2×2 with rank M = 1 = rank L is a selfadjoint
extension of Tmin if and only if TM,L = TP,Q :=
{
{˜z, ˜f} ∈ Tmax | P ˆz0 = 0 = Q ˆzN+1
}
,
where
P =
(
cos α0 sin α0
0 0
)
and Q =
(
0 0
− sin αN+1 cos αN+1
)
(7.25)
for a unique pair α0, αN+1 ∈ [0, π).
(ii) A linear relation TM,L given through M, L ∈ C2×2 with rank M = 2 = rank L is a selfadjoint
extension of Tmin if and only if TM,L = TR,β :=
{
{˜z, ˜f} ∈ Tmax | eiβR ˆz0 = ˆzN+1
}
with a unique β ∈ [0, π) and a symplectic matrix R ∈ R2×2.
– 126 –
7.2. Main results
Proof. Since the pairs of matrices P, Q and eiβR, I satisfy (7.23), Corollary 7.2.5 implies that
the linear relations TP,Q and TR,β are self-adjoint extensions of Tmin.
(i) Let TM,L be a linear relation given through M, L ∈ C2×2 satisfying (7.23) and with
rank M = 1 = rank L. Since by (1.4) we have dim[Ran M ∩ Ran L] = 0, it follows that
Mξ = Lη for some ξ, η ∈ C2 if and only if Mξ = 0 = Lη. Therefore the boundary conditions
in (7.24) can be expressed as M ˆz0 = 0 = L ˆzN+1. The rank condition implies that M = ab⊤
and L = cd⊤ for some vectors a, b, c, d ∈ C2K{0}. Then the equality MJM∗ = 0 = LJL∗ does
not depend on the vectors a, c and it is equivalent to b⊤Jb = 0 = d⊤Jd, which implies that b
and d are (scalar) complex multiples of vectors from R2. Thus, without loss of generality,
the vectors a, c may be chosen such that M, L can be written in the form as in (7.25) for
some α0, αN+1 ∈ [0, π). The uniqueness follows from the fact that cotan β = cotan γ with
β, ≥∈ (0, π) if and only if β = γ.
(ii) Finally, let TM,L be a linear relation given through M, L ∈ C2×2 satisfying (7.23)
and with rank M = 2 = rank L. Then the boundary conditions in (7.24) can be written as
ˆzN+1 = K ˆz0, where K := L−1M. Upon applying the second equality in (7.23) we obtain
that the matrix K is symplectic, i.e., KJK∗ = J. Therefore, K−1 = −JK∗J and | det K| = 1,
i.e., det K = eiε for some ε ∈ [0, 2π), which implies K−1 = e−iεKadj = −eiεJK⊤J, i.e.,
K∗⊤ = K = eiε K. If we put R := e−iε/2 K, i.e., K = eiε/2 R, then R = R and det R = 1, i.e.,
R ∈ R2×2 is a symplectic matrix. Uniqueness can be verified by a direct calculation. ■
As an illustration of the last theorem we provide a description of the Krein–von Neumann
extension of the minimal linear relation Tmin under Hypothesis 7.2.6.
Example 7.2.8. Assume that system (Sλ) is such that Hypothesis 7.2.6 holds and that the
minimal linear relation Tmin is positive, i.e., there exists c > 0 such that ⟨˜z, ˜f⟩ ≥ c|| ˜z||ψ for
all {˜z, ˜f} ∈ Tmin. Then the Krein–von Neumann self-adjoint extension extension of Tmin
admits the representation given in (A.14), i.e.,
TK = Tmin ∔ (Ker Tmax × {0}).
We show that TK can be also expressed as in the second part of Theorem 7.2.7 with
a suitable matrix R and a number β ∈ [0, 2π). By definition,
Ker Tmax =
{
˜z ∈ l2
ψ | {˜z, ˜0} ∈ Tmax
}
,
i.e., ˆz solves system (S0), i.e., L (ˆz)k = 0 on [0, N]Z. Because all solutions of system (S0) are
square summable in this case, the assumption of the definiteness of system (Sλ) implies
that dim Ker Tmax = 2. If ˜z ∈ dom TK, then there exist ˜v ∈ dom Tmin and ˜r ∈ Ker Tmax such
that ˜z = ˜v + ˜r or
ˆzk = ˆvk + ˆrk for all k ∈ [0, N + 1]Z, (7.26)
where ˆz ∈ ˜z, ˆv ∈ ˜v, and ˆr ∈ ˜r are the uniquely determined elements. Moreover, we have
ˆv0 = 0 = ˆvN+1 by (6.82) and ˆrk = α[1]
ˆr[1]
k
+ α[2]
ˆr[2]
k
for all k ∈ [0, N + 1]Z, where ˆr[1]
and ˆr[2]
form
a basis of Ker Tmax.
Let us define the matrix G =
(
a b
c d
)
:= (S0 × S1 × · · · × SN)−1 ∈ C2×2. Then one can
easily conclude that the matrix G is symplectic and every solution z ∈ C([0, N + 1]Z)2 of
system (S0) satisfies
zN+1 = Gz0. (7.27)
In the following construction we consider two cases: either b 0 or b = 0.
– 127 –
Chapter 7. Self-adjoint extensions
First, assume that b 0. Then there exist two solutions of system (S0) such that
ˆr[1]
0
=
(
0
1/b
)
and ˆr[2]
0
=
(
1
−a/b
)
.
These solutions are obviously linearly independent and by (7.27) we have
ˆr[1]
N+1
=
(
1
a/b
)
and ˆr[2]
N+1
=
(
0
c − da/b
)
.
If we take these two solutions as a basis of Ker Tmax, then (7.26) yields
ˆzk = ˆvk + α[1]
ˆr[1]
k
+ α[2]
ˆr[2]
k
for all k ∈ [0, N + 1]Z.
Upon evaluating ˆzk at k = 0 and k = N + 1 we obtain
ˆz0 =
(
α[2]
α[1]
/b − α[2]
a/b
)
and ˆzN+1 =
(
α[1]
α[1]
d/b + α[2]
c − α[2]
da/b
)
,
which for ˆzk =
( ˆxk
ˆuk
)
implies α[1]
= ˆxN+1 and α[2]
= ˆx0. Therefore
(
ˆxN+1
ˆxN+1 d/b + ˆx0 c − ˆx0 da/b
)
= ˆzN+1 = G ˆz0 = G
(
ˆx0
ˆxN+1/b − ˆx0 a/b
)
.
It means that ˆz ∈ dom TR,β, where β ∈ [0, π) is such that eiβ =
√
ad − bc, and R = e−iβ G, i.e.,
TK ⊆ TR,β. On the other hand, the linear relations TK and TR,β are self-adjoint extensions
of Tmin, thus TK = TR,β. Especially, if the coefficients a, b, c, d are real, then TR,β = TG,0.
If b = 0, then G =
(
a 0
c d
)
with |ad| = 1, i.e., d 0. In this case we proceed in the same
way with the basis of Ker Tmax given by the solutions ˜r[1]
and ˜r[2]
of (S0) such that
ˆr[1]
0
=
(
0
1/d
)
, ˆr[2]
0
=
(
1
−c/d
)
.
Then
( ˆx0 a
ˆuN+1
)
= ˆzN+1 = G ˆz0 = G
( ˆx0
ˆuN+1/d−ˆx0 c/d
)
. This shows (again) that TK = TR,β with
β ∈ [0, π) being such that eiβ =
√
ad, and R = e−iβ G.
In particular, let Sk =
(
1 −bk
0 1
)
and ψk =
(
wk 0
0 0
)
with bk > 0 and wk > 0 on [0, N]Z.
This system is definite on [0, N]Z and corresponds to the second order Sturm–Liouville
difference equation − [pk yk−1(λ)] = λwk yk(λ) with bk = 1/pk+1, see Example 6.2.4. Then
G =
(
1
∑N
k=0 bk
0 1
)
and by the previous part we have
TK =



{˜z, ˜f} ∈ Tmax | ˆz =
(
ˆx
ˆu
)
∈ C([0, N + 1]Z)2
, ˆu0 = ˆuN+1 =


N∑
k=0
bk


−1
× (ˆxN+1 − ˆx0)



.
▲
The boundary conditions in Theorem 7.2.7 include four particular cases. Namely,
with the notation ˆzk = (ˆxk, ˆuk)⊤ we get for α0 = 0 and αN+1 = π/2 the Dirichlet boundary
conditions ˆx0 = 0 = ˆxN+1, while for α0 = π/2 and αN+1 = 0 we have the Neumann
boundary conditions ˆu0 = 0 = ˆuN+1. The choice R = I and β = 0 yields the periodic
boundary conditions ˆz0 = ˆzN+1 and the choice R = I and β = π leads to the antiperiodic
boundary conditions ˆz0 = −ˆzN+1.
– 128 –
7.2. Main results
In the first part of the following theorem we show that any self-adjoint extension of
Tmin can be described by using the matrices determining the Dirichlet and Neumann
boundary conditions. For convenience, we introduce the general boundary trace map γM,L :
C(I+
Z )2 → C2 as
γM,L(ˆz) := M ˆz0 − LˆzN+1,
see also [38]. Then TM,L =
{
{˜z, ˜f} ∈ Tmax | γM,L(ˆz) = 0
}
. Especially, for P, Q given in (7.25)
we denote γx := γP,Q for α0 = 0, αN+1 = π/2, i.e., γx(ˆz) = 0 abbreviates the Dirichlet
boundary conditions, and similarly γu := γP,Q for α0 = π/2, αN+1 = 0, i.e., γu(ˆz) = 0
abbreviates the Neumann boundary conditions. In the second part of this theorem we
derive yet another equivalent representation of TM,L, which possesses the uniqueness
property.
Theorem 7.2.9. Let Hypothesis 7.2.6 be satisfied. Then the following hold.
(i) A linear relation T is a self-adjoint extension of Tmin if and only if there exist matrices
F, G ∈ C2×2 such that
rank(F, G) = 2, FG∗
= GF∗
(7.28)
and
T = TF,G :=
{
{˜z, ˜f} ∈ Tmax | Fγx(ˆz) + Gγu(ˆz) = 0
}
. (7.29)
(ii) We have TF,G = TF,G, where F, G satisfy (7.28), if and only if F = CF and G = CG for
some invertible matrix C ∈ C2×2.
(iii) A linear relation T is a self-adjoint extension of Tmin if and only if there exists a unitary
matrix V ∈ C2×2 such that
T = TV :=
{
{˜z, ˜f} ∈ Tmax | i(V − I)γx(ˆz) = (V + I)γu(ˆz)
}
. (7.30)
(iv) We have TV = TV, where V ∈ C2×2 is a unitary matrix, if and only if V = V.
Proof. (i) Let T be given by (7.29) with the matrices F, G ∈ C2×2 satisfying (7.28). If we
put M := FP0 + GPπ/2 and L := FQπ/2 + GQ0, where P0, Pπ/2 and Q0, Qπ/2 are the
matrices corresponding to P, Q defined in (7.25). Then MJM∗ −LJL∗ = FG∗ −GF∗ = 0 and
rank(F, G) = 2 is equivalent to rank(M, L) = 2. Hence M, L satisfy (7.23). Moreover, for
the left-hand side of the boundary conditions in (7.29) we have Fγx(ˆz) + Gγu(ˆz) = γM,L(ˆz).
Therefore {˜z, ˜f} ∈ TM,L if and only if {˜z, ˜f} ∈ TF,G, i.e., TF,G is a self-adjoint extension of Tmin
by Corollary 7.2.5. On the other hand, let T be a self-adjoint extension of Tmin, i.e., T = TM,L
with M, L ∈ C2×2 satisfying (7.23). If we put F := MP0 − LPπ/2 and G := LQ0 − MQπ/2,
then the conditions in (7.28) hold and γM,L(ˆz) can be written as in (7.29).
(ii) Sufficiency is clear. Assume that TF,G = TF,G for two pairs of matrices F, G and F, G
satisfying (7.28). Then, by (7.29), we have for any {˜z, ˜f} ∈ Tmax that Fγx(ˆz) + Gγu(ˆz) = 0
if and only if F γx(ˆz) + Gγu(ˆz) = 0. It means that ˆz0, ˆzN+1 solve simultaneously the both
systems of algebraic equations with the coefficient matrices F, G and F, G. It yields the
equivalence of systems, which implies an existence of an invertible matrix C ∈ C2×2 such
that F = CF and G = CG.
(iii) Let T be given by (7.30) with a unitary matrix V ∈ C2×2. If we put F := i
2(I − V)
and G := 1
2 (I + V). Then FG∗ = GF∗ and, by (1.2), rank(F, G) = 2, i.e., the matrices F, G
satisfy (7.28). Since the boundary conditions in (7.30) are equivalent to the boundary
conditions in (7.29) with F, G defined above, i.e., {˜z, ˜f} ∈ TF,G if and only if {˜z, ˜f} ∈ TV,
it follows from the previous part that the linear relation TV is a self-adjoint extension
– 129 –
Chapter 7. Self-adjoint extensions
of Tmin. On the other hand, let T be a self-adjoint extension of Tmin. Then, by the part
(i), we have T = TF,G with F, G ∈ C2×2 satisfying (7.28). Since by (1.2) and (7.28) we have
rank(F+iG) = 2, the matrix V := (F+iG)−1 (iG−F) is well-defined. One can directly verify
that V is a unitary matrix and the boundary conditions Fγx(ˆz) + Gγu(ˆz) = 0 are satisfied
if and only if i(V − I)γx(ˆz) − (V + I)γu(ˆz) = 0, i.e., TF,G = TV.
(iv) If V = V, then TV = TV. On the other hand, assume that TV = TV for two
unitary matrices V, V ∈ C2×2. Then TF,G = TV = TV = TF,G with the pairs of matrices
F, G and F, G being given as in the previous part. Then V = (F + iG)−1(iG − F) and
V = (F + iG)−1 (iG − F ) and by the part (ii) there exists an invertible matrix C ∈ C2×2 such
that F = CF and G = CG. Upon combining these facts we obtain V = V. ■
Remark 7.2.10.
(i) As a consequence of Theorem 7.2.7(i)-(ii) we obtain that TM,L = TM,L if and only if
M = CM and L = CL for some invertible matrix C ∈ C2×2.
(ii) Theorem 7.2.7(iii)-(iv) shows that the map from the set of all 2 × 2 unitary matrices
to the set of all self-adjoint extensions expressed as in (7.30) is a bijection.
7.3 Proof of main result
In this section, a proof is given for Theorem 7.2.1 which utilizes several arguments from
the linear algebra and whose main idea goes back to [162]. It is based on a construction
of a suitable GKN-set (see Theorem A.1) and on a more convenient expression than that
given in (7.15) for elements in dom Tmax. Similar results for system (2.6) can be found
in [135, Section 4].
Lemma 7.3.1. Let system (Sλ) be definite on the discrete interval IZ, {˜z, ˜f} ∈ Tmax be arbitrary,
and φ[1]
, . . . , φ[n+]
be arranged such that equality (7.18) holds. Then the element ˆz can be uniquely
expressed as
ˆzk = ˆvk +
2n∑
i=1
ηi ˆz[i]
k
+
p−2n∑
j=1
ζj φ
[j]
k
, k ∈ I+
Z , (7.31)
where ˆv ∈ dom Tmin, ˆz[1]
, . . . , ˆz[2n]
are specified in Lemma 6.4.3, and the numbers ηi, ζj ∈ C for all
i ∈ {1, . . . , 2n} and j ∈ {1, . . . ,p − 2n}. Moreover,
rank p−2n = p − 2n, (7.32)
where was defined in (7.14).
Proof. Since (7.18) is satisfied, there exists an invertible matrix P ∈ Cp×p such that
P =
(
Ip−2n 0(p−2n)×2n
Q R
)
, (7.33)
where 0(p−2n)×2n stands for the (p − 2n) × 2n zero matrix. If we put = ( [1]
, [2]
), where
[1]
∈ C2n×(p−2n) and [2]
∈ C2n×2n, and multiply (7.33) by from the left, we obtain
[1]
= − [2]
Q,
i.e., =
(
− [2]
Q, [2]
)
. It implies that rank [2]
= 2n by the second inequality in (1.3),
because rank = 2n. If we multiply equality (7.17) by ( [2]
)⊤−1 from the right, we get
Zk ( [2]
)⊤−1
= Vk ( [2]
)⊤−1
+ Θ[1]
k
[1]⊤
( [2]
)⊤−1
+ Θ[2]
k
,
– 130 –
7.3. Proof of main result
where Θ[1]
k
∈ C2n×(p−2n) and Θ[2]
k
∈ C2n×2n are such that (Θ+
k
, Θ−
k
) = (Θ[1]
k
, Θ[2]
k
) for all k ∈ I+
Z .
It shows that every solution φ[p−2n+1]
, . . . , φ[p]
can be uniquely expressed with ˆv[1]
, . . . , ˆv[2n]
,
ˆz[1]
, . . . , ˆz[2n]
, and φ[1]
, . . . , φ[p−2n]
, i.e.,
φ
[j]
k
= ˆr
[j]
k
+
2n∑
ℓ=1
ηj,ℓ ˆz[ℓ]
k
+
p−2n∑
s=1
ζj,s φ[s]
k
, k ∈ I+
Z , j ∈ {p − 2n + 1, . . . ,p}, (7.34)
for some ˆr[j]
∈ dom Tmin and ηj,ℓ, ζj,s ∈ C. Therefore the expression in (7.31) follows from
equality (7.15). Moreover, if we multiply both sides of (7.34) by φ[i]∗
k
J from the left, where
i ∈ {1, . . . ,p − 2n}, then
(φ[i]
, φ[j]
)N+1 = (φ[i]
, ˆv[j]
)N+1 +
2n∑
ℓ=1
ηj,ℓ (φ[i]
, ˆz[ℓ]
)N+1 +
p−2n∑
s=1
ζj,s (φ[i]
, φ[s]
)N+1.
Hence from (6.81) and the definition of ˆz[i]
we have
[1,2]
p−2n, n−
= p−2n T⊤
, (7.35)
where T ∈ Cn−×(p−2n) is a matrix consisting of the elements ζj,s for j ∈ {n+ + 1, . . . ,p} and
s ∈ {1, . . . ,p − 2n}. Since the solutions are arranged such that rank [1,2]
p−2n, n−
= p − 2n,
equality (7.32) follows from identity (7.35) and the second inequality in (1.3). ■
Remark 7.3.2. If we switch the role of s[·]
(λ0) and s[·]
(¯λ0) in the definition of φ[1]
, . . . , φ[p]
in
(7.13), i.e., we put φ[i]
= w[i]
(¯λ0) for i ∈ {1, . . . , n−} and φ[j+n−]
= v[j]
(λ0) for j ∈ {1, . . . , n+},
then the solutions φ[1]
, . . . , φ[n−]
can be arranged such that (7.31) and (7.32) hold.
Now we give the proof of Theorem 7.2.1.
Proof of Theorem 7.2.1. Assume that T is a self-adjoint extension of Tmin. Then, by Theorem
A.1, there exists a GKN-set {βj}
q
j=1
for (Tmin, Tmax) such that (A.12) holds. Since
βj ∈ Tmax, they may be identified as βj = {w[j]
, ˜h[j]
} ∈ Tmax. By Lemma 7.3.1, the elements
w[j]
can be uniquely expressed as
w
[j]
k
= ˆv
[j]
k
+
2n∑
i=1
ηj,i ˆz[i]
k
+
2q−2n∑
l=1
ζj,l φ[l]
k
, k ∈ I+
Z , (7.36)
where ˆv[j]
∈ dom Tmin and ηj,i, ζj,l ∈ C. We next show that the matrices
M :=
(
w[1]
0
, . . . , w
[q]
0
)∗
J ∈ Cq×2n
and L :=


ζ1,1 · · · ζ1,2q−2n
...
...
...
ζq,1 · · · ζq,2q−2n


∈ Cq×(2q−2n)
satisfy the relations in (7.19).
Since rank(M, L) ≤ q, let us assume that rank(M, L) < q. Then there exists a vector
c = (c1, . . . , cq)⊤ ∈ CqK{0} such that c∗(M, L) = 0, i.e., c∗ M = 0 = c∗ L. If wk :=
∑q
j=1
cjw
[j]
k
for k ∈ I+
Z , then we have w0 = JM∗ c = 0 and also (w, φ[i]
)N+1 =
∑q
j=1
cj (w[j]
, φ[i]
)N+1 for all
i ∈ {1, . . . , 2q − 2n}. Hence by (7.36) and (6.81) we have
(
(w, φ[1]
)N+1, . . . , (w, φ[2q−2n]
)N+1
)
= c∗
L 2q−2n = 0.
– 131 –
Chapter 7. Self-adjoint extensions
But then (w, ˆv)N+1 = 0 for any ˆv ∈ dom Tmax, because it can be written as in (7.31). It
means that w ∈ dom Tmin by (6.81) and hence β1, . . . , βq are linearly dependent in Tmax
modulo Tmin, which contradicts the assumption that that {βj}
q
j=1
is a GKN-set. Therefore,
the first condition in (7.19) is satisfied.
Next, we see that


(w[1]
, w[1]
)0 · · · (w[1]
, w[q]
)0
...
...
...
(w[q]
, w[1]
)0 . . . (w[q]
, w[q]
)0


= MJM∗
(7.37)
and by using (7.36), (6.81), and the definition of ˆz[i]
, also see that


(w[1]
, w[1]
)N+1 · · · (w[1]
, w[q]
)N+1
...
...
...
(w[q]
, w[1]
)N+1 . . . (w[q]
, w[q]
)N+1


= L 2q−2n L∗
. (7.38)
Since {βj}
q
j=1
is a GKN-set, we obtain from (6.67) that
0 = [βi : βj] = (w[i]
, w[j]
)k
N+1
0
for all i, j ∈ {1, . . . , q}. By (7.37) and (7.38), this implies that MJM∗ − L 2q−2n L∗ = 0, and so
the second condition in (7.19) is also satisfied.
For any ˆz ∈ dom Tmax, we can write


(w[1]
, ˆz)0
...
(w[q]
, ˆz)0


= M ˆz0 and


(w[1]
, ˆz)N+1
...
(w[q]
, ˆz)N+1


= L


(φ[1]
, ˆz)N+1
...
(φ[2q−2n]
, ˆz)N+1


, (7.39)
where the second equality follows from (7.36), (6.81), and the definition of ˆz[i]
. Upon
combining (A.12), (6.67), (7.39), we obtain that the linear relation T can be expressed as
T =
{
{˜z, ˜f} ∈ Tmax (ˆz, w[j]
)k
N+1
0
= 0 for all j ∈ {1, . . . , q}
}
=
{
{˜z, ˜f} ∈ Tmax w
[j]∗
k
J ˆzk
N+1
0
= 0 for all j ∈ {1, . . . , q}
}
=



{˜z, ˜f} ∈ Tmax M ˆz0 − L


(φ[1], ˆz)N+1
...
(φ[2q−2n], ˆz)N+1

 = 0



,
i.e., as written in (7.20).
On the other hand, let M ∈ Cq×2n and L ∈ Cq×(2q−2n) satisfy (7.19) and T be the linear
relation given by (7.20). We then must show that there exists a GKN-set {βj}
q
j=1
for
(Tmin, Tmax) such that T can be expressed as in (A.12). Denote the columns of JM∗ ∈ C2n×q
as ρ1, . . . , ρq and the columns of the matrix (φ[1]
k
, . . . , φ
[2q−2n]
k
)L∗ ∈ C2n×q as w[1]
k
, . . . , w
[q]
k
, i.e.,
ρi := JM∗
ei and w[i]
k
:=
2q−2n∑
l=1
ηi,l φ[l]
k
for all i ∈ {1, . . . , q}, (7.40)
– 132 –
7.3. Proof of main result
where ei is the i-th canonical unit vector in Cq and ηi,j are the elements of the matrix L
for i ∈ {1, . . . , q} and j ∈ {1, . . . , 2q − 2n}. Then w[i]
∈ Tmax for all i ∈ {1, . . . , q} and, by
Lemma 6.4.3, there exist βi := {˜r[i]
, ˜h[i]
} ∈ Tmax such that
ˆv[i]
0
= ρi and ˆv[i]
k
= w[i]
k
, k ∈ [b + 1, ∞)Z ∩ I+
Z
for all i ∈ {1, . . . , q}, where the number b is determined by the finite discrete interval
ID
Z := [a, b]Z ⊆ IZ with a, b ∈ IZ on which system (Sλ) is definite. We next show that {βi}
q
i=1
form a GKN-set for (Tmin, Tmax).
Since the linear independence of β1, . . . , βq in Tmax modulo Tmin is equivalent to the
linear independence of ˆv[1]
, . . . , ˆv[q]
in dom Tmax modulo Tmin, we assume that there exists
C = (c1, . . . , cq)⊤ ∈ CqK{0} such that
ˆv :=
q∑
j=1
cj ˆv[j]
∈ dom Tmin.
Then, from (6.81) and (7.40), we have for all φ[1]
, . . . , φ[2q−2n]
∈ Tmax that
0 =
(
(ˆv, φ[1]
)N+1, . . . , (ˆv, φ[2q−2n]
)N+1
)
= C∗
L 2q−2n.
This implies C∗ L = 0, because 2q−2n is assumed to be invertible, see (7.32). Simultaneously
we have ˆv0 = 0, which yields
0 = ˆv0 =
q∑
j=1
cj ˆv
[j]
0
= JM∗
C,
i.e., C∗ M = 0, because the matrix J is invertible. But this means C∗ (M, L) = 0, which
contradicts the first assumption in (7.19).
Next, let
Yk :=


(ˆv[1]
, ˆv[1]
)k · · · (ˆv[1]
, ˆv[q]
)k
...
...
...
(ˆv[q]
, ˆv[1]
)k · · · (ˆv[q]
, ˆv[q]
)k


.
Since it can be directly calculated that Y0 = MJM∗ and YN+1 = L 2q−2n L∗, the second
equality in (7.19) implies Y0 − YN+1 = 0. Therefore, by using (6.67), we get
[βi : βj] = (ˆv[i]
, ˆv[j]
)k
N+1
0
= 0,
which shows that {βi}
q
i=1
is a GKN-set for (Tmin, Tmax) as defined in the Appendix.
Finally, let {w, ˜g} ∈ Tmax be arbitrary, then
Mw0 =


(ˆv[1]
, w)0
...
(ˆv[q]
, w)0


and L


(φ[1]
, w)N+1
...
(φ[2q−2n]
, w)N+1


=


(ˆv[1]
, w)N+1
...
(ˆv[q]
, w)N+1


. (7.41)
By (6.67) the condition [{w, ˜g} : βi] = 0 is equivalent to
(w, ˆv[i]
)k
N+1
0
= 0 = −(ˆv[i]
, w)k
N+1
0
(7.42)
– 133 –
Chapter 7. Self-adjoint extensions
for all i ∈ {1, . . . , q}. Hence, by (7.41), we see that (7.42) can be written as
Mw0 − L


(φ[1]
, w)N+1
...
(φ[2q−2n]
, w)N+1


= 0.
Therefore the linear relation T given in (7.20) can be equivalently expressed as in (A.12),
which means that T is a self-adjoint extension of Tmin. ■
The simplification of Theorem 7.2.1 in the limit circle case is based on the following
lemma.
Lemma 7.3.3. Let system (Sλ) be definite on the discrete interval IZ and φ[1]
, . . . , φ[n+]
be arranged
as in Lemma 7.3.1. Assume that there exists a number ν ∈ R such that system (Sν) possesses
q := max{n+, n−} linearly independent square summable solutions (suppressing the argument ν)
given by θ[1]
, . . . , θ[q]
. Then these solutions can be arranged such that rank ϒp−2n = p−2n, where
ϒ :=


(θ[1]
, θ[1]
)N+1 . . . (θ[1]
, θ[q]
)N+1
...
...
...
(θ[q]
, θ[1]
)N+1 . . . (θ[q]
, θ[q]
)N+1


∈ Cq×q
.
Moreover, for any {˜z, ˜f} ∈ Tmax the element ˆz can be uniquely expressed as
ˆzk = ˆvk +
2n∑
i=1
αi ˆz[i]
k
+
p−2n∑
j=1
βj θ
[j]
k
, k ∈ I+
Z ,
where ˆv ∈ dom Tmin, ˆz[1]
, . . . , ˆz[2n]
are given in Lemma 6.4.3, and αi, βj ∈ C for all i ∈ {1, . . . , 2n}
and j ∈ {1, . . . ,p − 2n}.
Proof. Since θ[1]
, . . . , θ[q]
∈ dom Tmax, Lemma 7.3.1 implies that there exist unique numbers
αi,j, βi,ℓ ∈ C such that
θ[i]
k
= ˆv[i]
k
+
2n∑
j=1
αi,j ˆz
[j]
k
+
p−2n∑
ℓ=1
βi,ℓ φ[ℓ]
k
, k ∈ I+
Z , (7.43)
where i ∈ {1, . . . , q}. Then the definition of ˆz[i]
and identity (6.81) yield
ϒ = B p−2n B∗
, (7.44)
where the matrix B =
(
βi,j
)
∈ Cq×(p−2n). Hence rank ϒ ≤ p − 2n by the first inequality
in (1.3). On the other hand, by the Wronskian-type identity in (6.12) we have ϒ = ∗
0
J 0,
where k := (θ[1]
k
, . . . , θ
[q]
k
). Since the solutions θ[1]
k
, . . . , θ
[q]
k
are linearly independent, we
have rank k = q for all k ∈ I+
Z , and hence rank ϒ ≥ p−2n by the second inequality in (1.3).
Therefore rank ϒ = p − 2n, which implies that the solutions k := (θ[1]
k
, . . . , θ
[q]
k
) can be
arranged such that rank ϒp−2n = p − 2n. In this case, the invertibility of the submatrix
Bp−2n follows from the equality ϒp−2n = Bp−2n p−2n B∗
p−2n
, which is obtained analogously
to (7.44). Since from (7.43) we have
(θ[1]
k
, . . . , θ[p−2n]
k
) = (ˆv[1]
k
, . . . , ˆv[p−2n]
k
) + (ˆz[1]
k
, . . . , ˆz[2n]
k
)A∗
2n,p−2n + (φ[1]
k
, . . . , φ[p−2n]
k
)B∗
q−2n,
where A =
(
αi,j
)
∈ Cq×2n, the invertibility of Bp−2n means that φ[1]
k
, . . . , φ[p−2n]
k
can be
uniquely expressed by using θ[1]
k
, . . . , θ[p−2n]
k
, ˆv[1]
k
, . . . , ˆv[p−2n]
k
, and ˆz[1]
k
, . . . , ˆz[2n]
k
. Upon combining
these expressions with (7.31) we obtain the second part of the statement. ■
– 134 –
7.4. Bibliographical notes
7.4 Bibliographical notes
The results of this chapter were published in [A18]. Their generalization to symplectic
systems on time scales is one of the goals of the current research as well as an extension of
Theorem 7.1.1 to the discrete symplectic systems with B∗
k
Ck 0. The topic of the present
section is also closely related to the characterization of the spectrum of self-adjoint linear
relations, which represents another goal of our future research.
– 135 –
Chapter 7. Self-adjoint extensions
– 136 –
Appendix
Linear relations
... the theoryoflinearrelationwouldseemto havethepotential
for contributing to the enrichment and clarification of many
aspects of operator theory, including those concerned with
non-closable or non-densely-defined linear operators.
Ronald Cross, see [45, pg. iii]
In this supplementary chapter we recall several results from the theory of linear
relations, which are relevant to the content of Chapters 6 and 7. The theory of linear
relations has been established as a suitable tool for the study of multivalued or nondensely
defined linear operators in a Hilbert space. Its history goes back to [8] and the
results were further developed, e.g., in [42,45,46,82]. A (closed) linear relation T in a Hilbert
space H over the field of complex numbers C with the inner product ⟨·, ·⟩ is a (closed)
linear subspace of the product space H2 := H × H, i.e., the Hilbert space of all ordered
pairs {z, f } such that z, f ∈ H. The domain, range, kernel, and the multivalued part of T are
respectively defined as
dom T := {z ∈ H | {z, f } ∈ T }, (A.1)
Ran T :=
{
z ∈ H | there exists f ∈ H such that {z, f } ∈ T
}
, (A.2)
Ker T := {z ∈ H | {z, 0} ∈ T }, mul T :=
{
f ∈ H | {0, f } ∈ T
}
. (A.3)
In general, we let T (z) := {f ∈ H | {z, f } ∈ T }, and note that a linear relation T is the graph
of a linear operator in H when T (0) = {0}, i.e., when the subspace mul T is trivial. The
inverse of T , denoted as T −1, is the linear relation
T −1
:=
{
{f , z} | {z, f } ∈ T
}
and it satisfies
dom T −1
= Ran T , Ran T −1
= dom T , Ker T −1
= mul T , and mul T −1
= Ker T .
By T we mean the closure of T . The sum T + U and the algebraic sum T ∔ U are defined as
T + U :=
{
{z, f + g} | {z, f } ∈ T , {z, g} ∈ U
}
,
T ∔ U :=
{
{z + y, f + g} | {z, f } ∈ T , {y, g} ∈ U
}
.
– 137 –
Appendix
The adjoint T ∗ of the linear relation T is the closed linear relation given by
T ∗
:=
{
{y, g} ∈ H2
| ⟨z, g⟩ = ⟨f , y⟩ for all {z, f } ∈ T
}
. (A.4)
The definition of T ∗ reduces to the standard definition for the graph of the adjoint operator
when T is a densely defined operator. The adjoint linear relation T ∗ satisfies
T ∗
=
(
T
)∗
, T ∗∗
= T , Ker T ∗
= (Ran T )⊥
=
(
Ran T
)⊥
, (dom T )⊥
= mul T ∗
, (A.5)
A linear relation T is said to be symmetric (or Hermitian) if T ⊆ T ∗, and it is said to be
self-adjoint if T ∗ = T . It is easily seen that T is a symmetric linear relation if and only
if ⟨z, g⟩ = ⟨f , y⟩ for all {z, f }, {y, g} ∈ T. A symmetric linear relation T1 is said to be a
self-adjoint extension of T if T ⊆ T1 and T ∗
1
= T1.
For λ ∈ C and a linear relation T we define the linear relation
T − λI :=
{
{z, f − λz} ∈ H2
| {z, f } ∈ T
}
(A.6)
with the property (T − λI)∗ = T ∗ − ¯λI. Then
Mλ(T ) := Ker(T ∗
− λI) = {z ∈ H | {z, λz} ∈ T ∗
} (A.7)
is said to be the defect subspace of T and λ. Its dimension, i.e., the number
dλ(T ) := Ker(T ∗
− λI), (A.8)
is said to be the deficiency index of T and λ. Since
Ran(T − ¯λI)⊥
= Ker(T ∗
− λI)
the deficiency indices of T and T with the same λ are equal by (A.5), see [143, Lemma 2.4].
If T is a symmetric linear relation, the values of dλ(T ) are constant in the open upper
and lower half-planes of C, see [143, Theorem 2.13]. Hence we define the positive and
negative deficiency indices as d±(T ) := d±i(T ). If T is a closed symmetric linear relation,
then for every λ ∈ CKR the following direct sum decomposition (a generalization of the
von Neumann formula)
T ∗
= T ∔ Mλ(T ) ∔ M¯λ(T ) (A.9)
holds, where Mλ(T ) =
{
{z, λz} | {z, λz} ∈ T ∗
}
and the sum ∔ is orthogonal for λ = ±i,
see [116, Proposition 2.22]. A closed symmetric linear relation T possesses a self-adjoint
extension if and only if the positive and negative deficiency indices are equal, i.e., d+(T ) =
d−(T ), see [42, Corollary, pg. 34]. Moreover, it was shown in [116, Lemma 2.25] that
dλ(T) ≤ d±(T), (A.10)
whenever λ ∈ R and Ker(T − λI) = {0}.
Since the characterization of self-adjoint extensions of the minimal linear relation
associated with system (Sλ), see Chapter 7, is derived by applying the Glazman–Krein–
Naimark theory for linear relations, we recall the most fundamental parts of this theory,
see [143] for more details. A complex linear space S with a complex-valued function
– 138 –
Appendix
[:] : S × S → C is called pre-symplectic if it possesses the conjugate bilinear and skewHermitian
properties, i.e., for all P, Q, R ∈ S and α ∈ C we have
[P : Q + R] = [P : Q] + [P : R], [P + Q : R] = [P : R] + [Q : R],
[αP : Q] = α[P : Q], [P : αQ] = ¯α[P : Q],
[P : Q] = −[Q : P],
see [72] for more details. If we put S = H2 and
[{z, f } : {u, g}] := ⟨f , u⟩ − ⟨z, g⟩
for {z, f }, {u, g} ∈ H2, then S and [:] form the pre-symplectic space.
For a symmetric linear relation T ⊆ H2 we have
[T : T ] = 0 = [T : T ∗
] and T =
{
{z, f} ∈ T ∗
| [{z, f} : T ∗
] = 0
}
, (A.11)
see[143,Theorem3.5]. If, inaddition, the linear relationT isclosed and d := d+(T ) = d−(T ),
then the set {βj}d
j=1
with βj ∈ T ∗ for j ∈ {1, . . . , d} such that
1. β1, . . . , βd are linearly independent in T ∗ modulo T ,
2. [βj : βi] = 0 for all i, j ∈ {1, . . . , d},
is called GKN-set for the pair of linear relations (T , T ∗). The following theorem provides
the necessary and sufficient conditions for a linear relation T1 ⊆ H2 being a self-adjoint
extension of T , see [143, Theorem 4.7].
Theorem A.1. Let T ⊆ H2 be a closed symmetric linear relation such that d+(T ) = d−(T ) = d.
A subspace T1 ⊆ H2 is a self-adjoint extension of T if and only if there exists GKN-set {βj}d
j=1
for
(T , T ∗) such that
T1 =
{
F ∈ T ∗
| [F : βj] = 0 for all j = 1, . . . , d
}
. (A.12)
Finally, a linear relation T is called semibounded below, if there exists a ∈ R such that
⟨z, f ⟩ ≥ a⟨z, z⟩ for all {z, f } ∈ T . (A.13)
The number m(T ) := sup{a ∈ R | inequality (A.13) holds} is called the lower bound of T .
If m(T ) > 0, the linear relation T is said to be positive. Then, by analogy with the
case of densely defined positive symmetric operators (see [43, Theorem 5]), the smallest
and largest self-adjoint extensions of a positive symmetric linear relation are respectively
known as the Krein–von Neumann (or soft) extension TK and the Friedrichs (or hard) extension
TF. In particular, if T is closed and m(T ) > 0, then the Krein–von Neumann extension
admits the representation
TK = T ∔ (Ker T ∗
× {0}), (A.14)
see [43, Corollary 1] and also [84].
– 139 –
Appendix
– 140 –
List of symbols
We could, of course, use any notation we want; do not laugh at
notations; invent them, they are powerful. In fact, mathematics
is, to a large extent, invention of better notations.
Richard Phillips Feynman, see [78, pg. 17-7]
The items in the following list are sorted by their pronunciation or LATEX command. The
number refers to the page with the definition (or the first occurrence) of the symbol.
A
Madj (adjugate) . . . 3
B
S⊥ (orthogonal complement) . . . 4
C
Ck(λ) (Weyl circle for (Sλ)) . . . 74
Ck(λ) (Weyl circle for JVE) . . . 44
C (complex numbers) . . . 3
C(IZ)r×s, C(IZ)r (sequences over IZ) . . . 5
Cr×s (r × s matrices) . . . 3
Cr (r-dimensional vectors) . . . 3
C+ (upper half-plane of C) . . . 3
C− (lower half-plane of C) . . . 3
C0(IZ)r×s (compactly supported) . . . 5
Ck(λ) (Weyl circle for (Sλ)) . . . 20
codim S (codimension) . . . 4
CΨ (set for (Sλ)) . . . 73
CΨ,N (set for (Sλ)) . . . 73
D
D+(λ) (limiting Weyl disk for (Sλ)) . . . 75
Dk(λ) (Weyl disk for (Sλ)) . . . 74
D+(λ) (limiting Weyl disk for JVE) . . . 45
Dk(λ) (Weyl disk for JVE) . . . 44
δ(·) (sign of imaginary part) . . . 3
zk (forward difference) . . . 5
det M (determinant) . . . 3
diag{·} (diagonal matrix) . . . 3
dim Ran M (dimension of range) . . . 3
Dk(λ) (Weyl disk for (Sλ)) . . . 20
D+(λ) (limiting Weyl disk for (Sλ)) . . . 24
E
Ek(λ, M) (E(M)-function for (Sλ)) . . . 74
Ek(M) (E(M)-function for for JVE) . . . 44
Ek(M) (E(M)-function for (Sλ)) . . . 19
exp(M) (matrix exponential) . . . 5
Ek(M) (E(M)-function for (Sλ)) . . . 100
zk|n
m . . . 5
F
Fk(λ) (matrix for Weyl disk Dk(λ) ) . . . 21
– 141 –
List of symbols
G
Gk(λ) (matrix for Weyl disk Dk(λ)) . . . 74
Gk(λ) (matrix for Weyl disk Dk(λ)) . . . 44
(set for boundary conditions) . . . 16
Gk(λ) (matrix for Weyl disk Dk(λ)) . . . 21
Γ (boundary conditions for JVE) . . . 41
Gk,s(λ) (Green function for (Sλ)) . . . 101
H
Hk(λ) (matrix for Weyl disk Dk(λ)) . . . 74
Hk(λ) (matrix for Weyl disk Dk(λ)) . . . 44
Hk(λ) (matrix for Weyl disk Dk(λ)) . . . 21
H(t) (coefficient matrix for (2.5)) . . . 12
Hk (coefficient matrix for (2.6)) . . . 12
(H
R
λ ) (system for LC-invariance) . . . 55
(H
R
λ ) (system for LC-invariance) . . . 55
(Hλ) (system for LC-invariance) . . . 85
(Hλ) (system for LC-invariance) . . . 85
I
im(·) (imaginary part) . . . 3
⟨·, ·⟩ (semi-inner product for (Sλ)) . . . 26
⟨·, ·⟩Ψ(λ) (semi-inner product for (Sλ)) . . . 73
⟨·, ·⟩ ,N (finite semi-inner product) . . . 17
⟨·, ·⟩ψ (semi-inner product for (Sλ)) . . . 92
IZ, I+
Z (discrete interval) . . . 5
J
J . . . 5
K
Ker M (kernel) . . . 3
Kλ (map for ˜l2
ψ
and ˜l2
ψ,1
) . . . 110
L
Λk(λ, ¯ν) (matrix for (Sλ)) . . . 67
ℓ2
(space for (Sλ)) . . . 53
ℓ2 (space for (Sλ)) . . . 58
ℓ2 (space for (Sλ)) . . . 26
ℓ2
Ψ(λ)
(space for (Sλ)) . . . 75
ℓ2
W
(space with partial shift) . . . 76
L (z)k (natural map for (Sλ)) . . . 91
l2
ψ
(space for (Sλ)) . . . 92
l2
ψ,1
(space for (Sλ)) . . . 107
l2
ψ,0
(space for (Sλ)) . . . 107
˜l2
ψ
(Hilbert space for (Sλ)) . . . 106
˜l2×2
ψ
(space for (Sλ)) . . . 106
M
Mk(λ) (W-T function for JVE) . . . 43
Mk(λ) (W–T function) . . . 18
M+(λ) (half-line W–T function) . . . 25
N
N(λ) (ℓ2
Ψ(λ)
-solutions of (Sλ)) . . . 76
N(λ) (ℓ2
-solutions of (Sλ)) . . . 53
nλ (dimension of Nλ) . . . 115
Nλ (l2
ψ
-solutions of (Sλ)) . . . 115
N (natural numbers) . . . 3
N0 (natural numbers and zero) . . . 3
N(λ) (ℓ2 -solutions of (Sλ)) . . . 26
||·|| (semi-norm for (Sλ)) . . . 53
||·||2 (Euclidean vector norm) . . . 4
||·||1 (H¨older norm) . . . 4
||·|| (semi-norm for (Sλ)) . . . 26
||·||Ψ(λ) (semi-norm for (Sλ)) . . . 73
||·||σ (spectral norm) . . . 4
||·||ψ (semi-norm for (Sλ)) . . . 92
˜nλ (defect index for Tmin) . . . 115
Nλ (defect subspace for Tmin) . . . 115
kλ (space associated with Nλ) . . . 115
N0 (index for Hypothesis 2.3.7) . . . 22
N1 (index for Hypothesis 2.3.13) . . . 24
– 142 –
List of symbols
N2 (index for Hypothesis 2.3.15) . . . 25
N3 (index for Hypothesis 2.4.11) . . . 33
N4 (index for Hypothesis 5.3.2) . . . 74
O
M, ¯λ (conjugate) . . . 3
P
Pk(λ) (center of Dk(λ)) . . . 75
P+(λ) (center of D+(λ)) . . . 75
k(λ) (fundamental matrix for (Sλ)) . . . 73
Ψk(λ) (weight matrix for (Sλ)) . . . 69
P+(λ) (center of D+(λ)) . . . 45
Pk(λ) (center of Dk(λ)) . . . 45
k(λ) (fundamental matrix for (Sλ)) . . . 49
k (weight matrix for (Sλ)) . . . 49
Φk(λ) (fundamental matrix for JVE) . . . 42
k (weight matrix for (Sλ)) . . . 55
k(λ) (fundamental matrix for (Sλ)) . . . 16
π(z) (quotient space map for ˜l2
ψ
) . . . 106
Pk(λ) (center of Dk(λ)) . . . 22
P+(λ) (center of D+(λ)) . . . 23
k (weight matrix for (Sλ)) . . . 11
k (weight matrix for (Sλ)) . . . 57
ψk (weight matrix for (Sλ)) . . . 89
M > 0 (positive definite) . . . 3
M ≥ 0 (positive semidefinite) . . . 3
z[s]
k
(λ) (partial shift) . . . 71
Q
(Qλ) (system for LC-invariance) . . . 80
(Qλ) (system for LC-invariance) . . . 79
(Qλ) (system for LC-invariance) . . . 80
R
Rk(λ) (radius of Dk(λ)) . . . 75
R+(λ) (radius of D+(λ)) . . . 75
R+(λ) (radius of D+(λ)) . . . 45
Rk(λ) (radius of Dk(λ)) . . . 45
Ran M (range) . . . 3
rank M (rank) . . . 3
R (real numbers) . . . 3
re(·) (real part) . . . 3
Rk(λ) (radius of Dk(λ)) . . . 22
r(λ) (rank of R+(λ)) . . . 28
R+(λ) (radius of D+(λ)) . . . 23
S
Sk(λ) (coefficient matrix for (Sλ)) . . . 65
Sk (coefficient matrix for (Sλ)) . . . 49
Sk (coefficient matrix for (Sλ)) . . . 55
5k(λ) (coefficient matrix for (Sλ)) . . . 11
Sk (coefficient matrix for (Sλ)) . . . 11
S[·]
k
(coefficient matrix for (Sλ)) . . . 65
sprad M (spectral radius) . . . 3
pk(λ) (coefficient matrix for (Sλ)) . . . 89
Sk (coefficient matrix for (Sλ)) . . . 89
Sk (coefficient matrix for (Sλ)) . . . 55
M∗, M∗(·) (conjugate transpose) . . . 3
Mp,q (p × q submatrix) . . . 3
Mp (p × p submatrix) . . . 3
(Sλ) (system linear in λ) . . . 11
(Sλ) (augmented system (Sλ)) . . . 49
(Eλ) (equation for LC-invariance) . . . 62
(Eλ) (equation for LC-invariance) . . . 62
(Sλ) (system for LC-invariance) . . . 55
(Sλ) (system for LC-invariance) . . . 55
(Sλ) (system analytic in λ) . . . 65
(S
f
λ
) (nonhomogeneous system) . . . 92
(Sλ) (time-reversed version of (Sλ)) . . . 89
T
T k(λ) (shift matrix for (Qλ)) . . . 82
– 143 –
List of symbols
Tmax (maximal linear relation) . . . 107
Tmin (minimal linear relation) . . . 113
TM,L (self-adjoint extension of Tmin) . . . 126
T0 (pre-minimal linear relation) . . . 107
TP,Q (self-adjoint extension of Tmin) . . . 126
Tk(λ) (shift matrix for (5.17)) . . . 71
tr M (trace) . . . 3
TR,β (self-adjoint extension of Tmin) . . . 126
Θk(λ) (fundamental matrix for (Sλ)) . . . 93
T k(λ) (shift matrix for (Qλ)) . . . 82
ϑ(λ, IZ) (matrix for (Sλ)) . . . 98
M⊤, M⊤(·) (transpose) . . . 3
TK (K–vN extension of Tmin) . . . 127
U
777 (set of 4n × 4n unitary matrices) . . . 45
U (set of 2n × 2n unitary matrices) . . . 22
V
888 (set of 4n×4n contractive matrices) . . . 45
Vk (coefficient matrix for (Sλ)) . . . 49
Vk (coefficient matrix for (Sλ)) . . . 55
Vk (coefficient matrix for (Sλ)) . . . 89
Vk (coefficient matrix for (Sλ)) . . . 55
V (set of 2n×2n contractive matrices) . . . 22
Vk (coefficient matrix for (Sλ)) . . . 11
W
Wk (weight matrix for (5.17)) . . . 71
W(t) (weight matrix for (2.5)) . . . 12
Wk (weight matrix for (2.6)) . . . 12
X
Xk(λ, M) (Weyl solution for (Sλ)) . . . 74
Xk(λ) (Weyl solution for JVE) . . . 43
Xk(λ) (Weyl solution for (Sλ)) . . . 100
Xk(λ) (Weyl solution for (Sλ)) . . . 17
Z
Zk(λ) (second half of k(λ)) . . . 73
Zk(λ) (second half of k(λ)) . . . 51
Zk(λ) (second half of Θk(λ)) . . . 100
Zk(λ) (second half of k(λ)) . . . 16
Z (integers) . . . 3
– 144 –
List of author’s publications (as of August 22, 2016)
Every mathematical discipline goes through three periods
of development: the naive, the formal, and the critical.
David Hilbert, see [131, p. 240]
Items [A1, A2, A4–A21] are indexed in the MathSciNet Database. Moreover, items
[A1,A6–A9] were included in the author’s doctoral dissertation [A24].
2009
[A1] R. Hilscher and P. Zem´anek, Trigonometric and hyperbolic systems on time scales,
Dynam. Systems Appl. 18 (2009), no. 3-4, 483–505. (Cited on page 145.)
[A2] R. ˇSimon Hilscher and P. Zem´anek, Definiteness of quadratic functionals for Hamiltonian
and symplectic systems: a survey, Int. J. Difference Equ. 4 (2009), no. 1, 49–67.
(Cited on page 145.)
[A3] P. Zem´anek, Discrete trigonometric and hyperbolic systems: An overview, Ulmer Seminare
¨uber Funktionalanalysis und Differentialgleichungen 14 (2009), 345–359.
(Cited on page 6.)
2010
[A4] S. L. Clark and P. Zem´anek, On a Weyl–Titchmarsh theory for discrete symplectic
systems on a half line, Appl. Math. Comput. 217 (2010), no. 7, 2952–2976. (Cited on
pages 1, 4, 13, 16, 18, 19, 22, 25, 26, 27, 33, 38, 41, 50, 100, 101, 102, 105, and 145.)
[A5] R. ˇSimon Hilscher and P. Zem´anek, Friedrichs extension of operators defined by linear
Hamiltonian systems on unbounded interval, Math. Bohem. 135 (2010), no. 2, 209–222.
(Cited on pages 119 and 145.)
2011
[A6] P. Hasil and P. Zem´anek, Critical second order operators on time scales, Discrete Contin.
Dyn. Syst. 2011 (2011), suppl., 653–659. (Cited on page 145.)
– 145 –
List of author’s publications
[A7] R. ˇSimon Hilscher and P. Zem´anek, Weyl–Titchmarsh theory for time scale symplectic
systems on half line, Abstr. Appl. Anal. 2011 (2011), Art. ID 738520, 41 pp. (electronic).
(Cited on pages 14 and 145.)
[A8] R. ˇSimon Hilscher and P. Zem´anek, Overview of Weyl–Titchmarsh theory for second
order Sturm–Liouville equations on time scales, Int. J. Difference Equ. 6 (2011), no. 1,
39–51. (Cited on page 145.)
[A9] P. Zem´anek, Krein-von Neumann and Friedrichs extensions for second order operators
on time scales, Int. J. Dyn. Syst. Differ. Equ. 3 (2011), no. 1–2, 132–144. (Cited on
page 145.)
2012
[A10] R. ˇSimon Hilscher and P. Zem´anek, New results for time reversed symplectic dynamic
systems and quadratic functionals, Electron. J. Qual. Theory Differ. Equ. (2012), no. 15,
11 pp. (electronic). (Cited on page 145.)
[A11] P. Zem´anek, Rofe–Beketov formula for symplectic systems, Adv. Difference Equ. 2012
(2012), no. 104, 9 pp. (electronic). (Cited on page 145.)
[A12] P. Zem´anek and P. Hasil, Friedrichs extension of operators defined by Sturm–Liouville
equations of higher order on time scales, Appl. Math. Comput. 218 (2012), no. 22,
10829–10842. (Cited on page 145.)
2013
[A13] R. ˇSimon Hilscher and P. Zem´anek, Weyl disks and square summable solutions for
discrete symplectic systems with jointly varying endpoints, Adv. Difference Equ. 2013
(2013), no. 232, 18 pp. (electronic). (Cited on pages 1, 2, 54, and 145.)
[A14] P. Zem´anek, A note on the equivalence between even-order Sturm–Liouville equations and
symplectic systems on time scales, Appl. Math. Lett. 26 (2013), no. 1, 134–139. (Cited
on page 145.)
2014
[A15] R. ˇSimon Hilscher and P. Zem´anek, Weyl–Titchmarsh theory for discrete symplectic
systems with general linear dependence on spectral parameter, J. Difference Equ. Appl. 20
(2014), no. 1, 84–117. (Cited on pages 1, 39, and 145.)
[A16] R. ˇSimon Hilscher and P. Zem´anek, Limit point and limit circle classification for symplectic
systems on time scales, Appl. Math. Comput. 233 (2014), 623–646. (Cited on
pages 1, 2, 14, 39, 54, and 145.)
[A17] R. ˇSimon Hilscher and P. Zem´anek, Generalized Lagrange identity for discrete symplectic
systems and applications in Weyl–Titchmarsh theory, in “Theory and Applications
of Difference Equations and Discrete Dynamical Systems”, Proceedings of the 19th
International Conference on Difference Equations and Applications (Muscat, 2013),
– 146 –
List of author’s publications
Z. AlSharawi, J. Cushing, and S. Elaydi (editors), Springer Proceedings in Mathematics
& Statistics, Vol. 102, pp. 187–202, Springer, Berlin, 2014. (Cited on pages 1,
2, 87, 90, and 145.)
2015
[A18] S. L. Clark and P. Zem´anek, On discrete symplectic systems: Associated maximal and
minimal linear relations and nonhomogeneous problems, J. Math. Anal. Appl. 421 (2015),
no. 1, 779–805. (Cited on pages 1, 2, 89, 117, 135, and 145.)
[A19] R. ˇSimon Hilscher and P. Zem´anek, Limit circle invariance for two differential systems
on time scales, Math. Nachr. 288 (2015), no. 5-6, 696–709. (Cited on pages 1, 2, 63,
and 145.)
[A20] R. ˇSimon Hilscher and P. Zem´anek, Time scale symplectic systems with analytic dependence
on spectral parameter, J. Difference Equ. Appl. 21 (2015), no. 3, 209–239. (Cited
on pages 1, 2, 73, 87, and 145.)
2016
[A21] P. Zem´anek and S. L. Clark, Characterization of self-adjoint extensions for discrete
symplectic systems, J. Math. Anal. Appl. 440 (2016), no. 1, 323–350. (Cited on pages 1,
2, 89, 117, and 145.)
Accepted
[A22] P. Zem´anek, Limit point criteria for second order Sturm–Liouville equations on time scales,
in “Differential and Difference Equations with Applications”, Proceedings of the
International Conference on Differential & Difference Equations and Applications
2015 (Amadora, 2015), S. Pinelas, O. Doˇsl´y, Z. Doˇsl´a, and P. Kloeden (editors),
Springer Proceedings in Mathematics & Statistics, Springer, Berlin, to appear. (Not
cited.)
Submitted
[A23] P. Zem´anek, Principal solution in Weyl–Titchmarsh theory for second order Sturm–
Liouville equation on time scales, submitted. (Cited on pages 36 and 123.)
Doctoral dissertation
[A24] P. Zem´anek, New Results in Theory of Symplectic Systems on Time Scales, doctoral
dissertation, Masaryk University, Brno, 2011. ISBN 978-80-210-5515-5. (Cited on
page 145.)
– 147 –
List of author’s publications
– 148 –
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