Spectral and pseudospectral functions of various dimensions for symmetric systems
The main object of the paper is a symmetric system Jy′ − B(t)y = λ∆(t)y defined on an interval I = [a, b) with the regular endpoint a.
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irk-123456789-1409022018-07-18T01:23:56Z Spectral and pseudospectral functions of various dimensions for symmetric systems Mogilevskii, V.I. The main object of the paper is a symmetric system Jy′ − B(t)y = λ∆(t)y defined on an interval I = [a, b) with the regular endpoint a. 2016 Article Spectral and pseudospectral functions of various dimensions for symmetric systems / V.I. Mogilevskii // Український математичний вісник. — 2016. — Т. 13, № 2. — С. 224-269. — Бібліогр.: 33 назв. — рос. 1810-3200 2010 MSC: 34B08,34B20,34B40,34L10,47A06 http://dspace.nbuv.gov.ua/handle/123456789/140902 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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The main object of the paper is a symmetric system Jy′ − B(t)y = λ∆(t)y defined on an interval I = [a, b) with the regular endpoint a. |
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Mogilevskii, V.I. |
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Mogilevskii, V.I. Spectral and pseudospectral functions of various dimensions for symmetric systems Український математичний вісник |
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Mogilevskii, V.I. |
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Mogilevskii, V.I. |
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Spectral and pseudospectral functions of various dimensions for symmetric systems |
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Spectral and pseudospectral functions of various dimensions for symmetric systems |
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Spectral and pseudospectral functions of various dimensions for symmetric systems |
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Spectral and pseudospectral functions of various dimensions for symmetric systems |
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Spectral and pseudospectral functions of various dimensions for symmetric systems |
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spectral and pseudospectral functions of various dimensions for symmetric systems |
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Інститут прикладної математики і механіки НАН України |
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Spectral and pseudospectral functions of various dimensions for symmetric systems / V.I. Mogilevskii // Український математичний вісник. — 2016. — Т. 13, № 2. — С. 224-269. — Бібліогр.: 33 назв. — рос. |
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Український математичний вісник |
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AT mogilevskiivi spectralandpseudospectralfunctionsofvariousdimensionsforsymmetricsystems |
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2025-07-10T11:30:42Z |
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1837259339680710656 |
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Український математичний вiсник
Том 13 (2016), № 2, 224 – 269
Spectral and pseudospectral functions of various
dimensions for symmetric systems
Vadim Mogilevskii
(Presented by M.M. Malamud)
Abstract. The main object of the paper is a symmetric system
Jy′
− B(t)y = λ∆(t)y defined on an interval I = [a, b) with the reg-
ular endpoint a. Let ϕ(·, λ) be a matrix solution ϕ(·, λ) of this sys-
tem of an arbitrary dimension and let (V f)(s) =
∫
I
ϕ∗(t, s)∆(t)f(t) dt
be the Fourier transform of the function f(·) ∈ L2
∆(I). We define a
pseudospectral function of the system as a matrix-valued distribution
function σ(·) of the dimension nσ such that V is a partial isometry from
L2
∆(I) to L2(σ;Cnσ ) with the minimally possible kernel. Moreover, we
find the minimally possible value of nσ and parameterize all spectral
and pseudospectral functions of every possible dimensions nσ by means
of a Nevanlinna boundary parameter. The obtained results develop the
results by Arov and Dym; A. Sakhnovich, L. Sakhnovich and Roitberg;
Langer and Textorius.
2010 MSC. 34B08, 34B20, 34B40, 34L10, 47A06.
Key words and phrases. Symmetric differential system, pseudospec-
tral function, Fourier transform, m-function, dimension of a spectral
function.
1. Introduction
Let H and Ĥ be finite dimensional Hilbert spaces, let H = H⊕Ĥ⊕H
and let [H] be the set of linear operators in [H]. Recall that a non-
decreasing left continuous operator (matrix) function σ(·) : R → [H] with
σ(0) = 0 is called a distribution function of the dimension nσ := dimH.
We consider symmetric differential system [3,11]
Jy′ −B(t)y = λ∆(t)y, t ∈ I, λ ∈ C, (1.1)
Received 26.05.2016
ISSN 1810 – 3200. c© Iнститут математики НАН України
V. Mogilevskii 225
where B(t) = B∗(t) and ∆(t) ≥ 0 are [H]-valued functions defined on an
interval I = [a, b), b ≤ ∞, and integrable on each compact subinterval
[a, β] ⊂ I and
J =
0 0 −IH
0 iI
Ĥ
0
IH 0 0
: H ⊕ Ĥ ⊕H → H ⊕ Ĥ ⊕H. (1.2)
System (1.1) is called a Hamiltonian system if Ĥ = {0} and hence H =
H ⊕H,
J =
(
0 −IH
IH 0
)
: H ⊕H → H ⊕H.
The system is called regular if b <∞ and
∫
I ||B(t)||dt <∞,
∫
I ||∆(t)||dt <
∞.
As is known a spectral function is a basic concept in the theory of
eigenfunction expansions of differential operators (see e.g. [10, 29] and
references therein). In the case of a symmetric system definition of the
spectral function requires a certain modification. Namely, let H = L2
∆(I)
be the Hilbert space of functions f : I → H satisfying∫
I
(∆(t)f(t), f(t)) dt <∞.
Assume that system (1.1) is Hamiltonian and ϕ(·, λ) is an [H,H ⊕ H]-
valued operator solution of (1.1) such that ϕ(0, λ) = (0, IH)⊤. An [H]-
valued distribution function σ(·) is called a spectral function of the system
if the (generalized) Fourier transform Vσ : H → L2(σ;H) defined by
(Vσf)(s) = f̂0(s) :=
∫
I
ϕ∗(t, s)∆(t)f(t) dt, f(·) ∈ H (1.3)
is an isometry. If σ(·) is a spectral function, then the inverse Fourier
transform is defined for each f ∈ H by
f(t) =
∫
I
ϕ(t, s) dσ(s)f̂0(s) (1.4)
(the integrals in (1.3) and (1.4) converge in the norm of L2(σ;H) and H
respectively). If the operator ∆(t) is invertible a.e. on I, then spectral
functions exist. Otherwise the Fourier transform may have a nontrivial
kernel kerVσ and hence the set of spectral functions may be empty. The
natural generalization of a spectral function to this case is an [H]-valued
distribution function σ(·) such that the Fourier transform Vσ of the form
(1.3) is a partial isometry. If σ(·) is such a function, then the inverse
Fourier transform (1.4) is valid for each f ∈ H ⊖ kerVσ. Therefore an
226 Pseudospectral functions
interesting problem is a characterization of [H]-valued distribution func-
tions σ(·) such that the Fourier transform Vσ is a partial isometry with
the minimally possible kernel kerVσ. This problem was solved in [2,31,32]
for regular systems and in [27] for general systems. The results of [27] was
obtained in the framework of the extension theory of symmetric linear
relations. As is known [14,18, 22, 30] system (1.1) generates the minimal
(symmetric) linear relation Tmin and the maximal relation Tmax(= T ∗
min)
in H. Let T ⊃ Tmin be a symmetric relation in H given by
T = {{y, f} ∈ Tmax : (IH , 0)y(a) = 0
and lim
t→b
(Jy(t), z(t)) = 0, z ∈ domTmax}
and let mulT be the multivalued part of T . It was shown in [27] that
for each [H]-valued distribution function σ(·) such that Vσ is a partial
isometry the inclusion mulT ⊂ kerVσ is valid. This fact makes natural
the following definition.
Definition 1.1. [27] An [H]-valued distribution function σ(·) is called a
pseudospectral function of the system (1.1) (with respect to ϕ(·, λ)) if the
Fourier transform Vσ is a partial isometry with the minimally possible
kernel kerVσ = mulT .
If the Hamiltonian system is regular, then kerVσ = {f ∈ H : f̂0(s) =
0, s ∈ R} and therefore Definition 1.1 turns into the definition of the
pseudospectral function from the monographes [2, 32]. In these mono-
graphes all [H]-valued pseudospectral functions of the regular system are
parameterized in the form of a linear fractional transform of a Nevan-
linna parameter. Similar result for singular systems was obtained in our
paper [27]. Observe also that an existence of [H]-valued pseudospectral
functions of the singular Hamiltonian system in the case dimH = 1 was
proved in [14].
Assume now that system (1.1) is not necessarily Hamiltonian. Let
Y (·, λ) be the [H]-valued operator solution of (1.1) with Y (a, λ) = IH
and let Σ(·) be an [H]-valued distribution function such that the Fourier
transform VΣ : H → L2(Σ;H) defined by
(VΣf)(s) = f̂(s) :=
∫
I
Y ∗(t, s)∆(t)f(t) dt, f(·) ∈ H (1.5)
is a partial isometry. Moreover, let mulTmin be the multivalued part of
Tmin. Then according to [26] mulTmin ⊂ kerVΣ and the same arguments
as for transform (1.3) make natural the following definition.
V. Mogilevskii 227
Definition 1.2. [26] An [H]-valued distribution function Σ(·) is called a
pseudospectral function of the system (1.1) (with respect to Y (·, λ)) if the
Fourier transform VΣ is a partial isometry with the minimally possible
kernel kerVΣ = mulTmin.
Existence of pseudospectral functions Σ(·) follows from the results
of [8,9,16,17]. In [16,17] a parametrization of all pseudospectral functions
Σ(·) of the regular system (1.1) is given. This parametrization is closed
to that of the [H]-valued pseudospectral functions σ(·) in [2,32]. Similar
result for singular systems is obtained in [26].
In the present paper we continue our investigations of pseudospectral
and spectral functions of symmetric systems contained in [1, 26,27].
According to Definitions 1.1 and 1.2 the dimensions of pseudospectral
functions Σ(·) and σ(·) are nΣ = dimH and nσ = dimH(< nΣ). In this
connection the following problems seems to be interesting:
• To define naturally a spectral and pseudospectral function of an
arbitrary dimension for the system (1.1) and describe all such functions
by analogy with [26,27].
• To characterize spectral functions of the minimally possible dimension
The paper is devoted to the solution of these problems.
Let H0 and θ be subspaces in H, Kθ ∈ [H0,H] be an operator isomor-
phically mapping H0 onto θ and ϕ(·, λ) be the [H0,H]-valued operator
solution of (1.1) with ϕ(a, λ) = Kθ. Moreover, let σ(·) be an [H0]-valued
distribution function such that the Fourier transform Vσ : H → L2(σ;H0)
defined by (1.3) is a partial isometry. It turns out that mulT ⊂ kerVσ,
where mulT is the multivalued part of a symmetric relation T ⊃ Tmin in
H given by
T = {{y, f} ∈ Tmax : y(a) ∈ θ and lim
t→b
(Jy(t), z(t)) = 0, z ∈ domTmax}.
This statement makes natural the following most general definition of
pseudospectral and spectral functions.
Definition 1.3. An [H0]-valued distribution function σ(·) is called a
pseudospectral function of the system (1.1) (with respect to the operator
Kθ) if the Fourier transform Vσ is a partial isometry with the minimally
possible kernel kerVσ = mulT .
A pseudospectral function σ(·) with kerVσ = {0} is called a spectral
function.
It turns out that actually a pseudospectral function with respect to
the operator Kθ is uniquely characterized by the subspace θ ⊂ H.
228 Pseudospectral functions
We parametrize all pseudospectral (spectral) functions for a given θ
and find a lower bound of the dimension of all spectral functions σ(·)
corresponding to various θ. More precisely the following three theorems
are the main results of the paper.
Theorem 1.4. Assume that system (1.1) is definite (see Definition 3.15)
and deficiency indices n±(Tmin) of Tmin satisfy n−(Tmin) ≤ n+(Tmin).
Moreover, let θ be a subspace in H and let θ× := H ⊖ Jθ. Then a
pseudospectral function σ(·) (with respect to Kθ) exists if and only if
θ× ⊂ θ.
Theorem 1.5. Assume that θ is a subspace in H such that θ× ⊂ θ and
there exists only a trivial solution y = 0 of the system (1.1) such that
∆(t)y(t) = 0 (a.e. on I) and y(a) ∈ θ (the last condition is fulfilled
for definite systems). Moreover, let for simplicity n+(Tmin) = n−(Tmin).
Then:
(1) There exist auxiliary finite-dimensional Hilbert spaces H0 ⊂ H
and Ḣ, an operator U = Uθ ∈ [H0,H] isomorphically mapping H0 onto
θ, Nevanlinna operator functions m0(λ)(∈ [H0]), Ṁ(λ)(∈ [Ḣ]) and an
operator function S(λ)(∈ [Ḣ,H0]) such that the equalities
mτ (λ) = m0(λ) + S(λ)(C0(λ)− C1(λ)Ṁ(λ))−1C1(λ)S
∗(λ), λ ∈ C \ R
(1.6)
στ (s) = lim
δ→+0
lim
ε→+0
1
π
∫ s−δ
−δ
Immτ (u+ iε) du (1.7)
establish a bijective correspondence σ(s) = στ (s) between all Nevanlinna
operator pairs τ = {C0(λ), C1(λ)}, Cj(λ) ∈ [Ḣ], j ∈ {0, 1}, satisfying
the admissibility conditions
lim
y→∞
1
iy (C0(iy)− C1(iy)Ṁ(iy))−1C1(iy) = 0 (1.8)
lim
y→∞
1
iyṀ(iy)(C0(iy)− C1(iy)Ṁ(iy))−1C0(iy) = 0 (1.9)
and all pseudospectral functions σ(·) of the system (with respect to U).
Moreover, each pair τ is admissible (and hence the conditions (1.8) and
(1.9) may be omitted) if and only if lim
y→∞
1
iyṀ(iy) = 0 and lim
y→∞
y ·
Im(Ṁ(iy)h, h) = +∞, 0 6= h ∈ Ḣ.
(2) The set of spectral functions (with respect to U) is not empty
if and only if mulT = {0}. If this condition id fulfilled, then the sets
of spectral and spectral function (with respect to U) coincide and hence
statement (1) holds for spectral functions.
V. Mogilevskii 229
Theorem 1.6. Let system (1.1) be definite and let n−(Tmin) ≤ n+(Tmin).
Then the set of spectral functions of the system is not empty if and only
if mulTmin = {0}. If this condition is fulfilled, then the dimension nσ
of each spectral function σ(·) satisfies dim(H ⊕ Ĥ) ≤ nσ ≤ dimH and
there exists a subspace θ ⊂ H and a spectral function σ(·) (with respect to
Kθ) such that the dimension nσ of σ(·) has the minimally possible value
nσ = dim(H ⊕ Ĥ).
Note that the coefficients m0(λ), S(λ) and Ṁ(λ) in (1.6) are defined
in terms of the boundary values of respective operator solutions of (1.1)
at the endpoints a and b. Observe also that mτ (λ) in (1.6) is an [H0]-
valued Nevanlinna function (the m-function of the system) and (1.7) is
the Stieltjes formula for mτ (·). If the system is Hamiltonian, θ is a
self-adjoint linear relation in H ⊕ H and τ = τ∗, then mτ (λ) is the
Titchmarsh–Weyl function of the system corresponding to self-adjoint
separated boundary conditions [13]. In the case of a non-Hamiltonian
system such conditions do not exist [22] and mτ (λ) corresponds to special
mixed boundary conditions (see Definition 4.16).
For pseudospectral functions σ(·) of the minimal dimension nσ =
dim(H ⊕ Ĥ) formulas similar to (1.6) and (1.7) were obtained in [1].
These formulas are proved in [1] only for a parameter τ of a special form;
therefore not all pseudospectral functions σ(·) are parametrize in this
paper.
As is known [15, 28] the set of spectral functions of a symmetric dif-
ferential operator l[y] of an order m coincides with the set of spectral
functions of a special definite symmetric system corresponding to l[y].
Moreover, this system is Hamiltonian if and only if m is even. According
to the classical monograph by N. Dunford and J. T. Schwartz [10, ch.
13.21] an important problem of the spectral theory of differential oper-
ators is a characterization of their spectral functions σmin(·) with the
minimally possible dimension nmin. It follows from Theorem 1.6 that
nmin = k + 1 in the case m = 2k + 1 and nmin = k in the case m = 2k.
Moreover, by using Theorem 1.5 one may obtain a parametrization of
σmin(·). In more details this results will be specified elsewhere.
For a differential operator l[y] of an even order m formulas similar to
(1.6) and (1.7) were proved in our paper [23]. These formulas enable one
to calculate spectral functions σ(·) of an arbitrary dimension nσ (m2 ≤
nσ ≤ m) corresponding to a special parameter τ ; hence they do not
parametrize all spectral functions of l[y].
In conclusion note that our approach is based on the theory of bound-
ary triplets (boundary pairs) for symmetric linear relations and their Weyl
function (see [4, 6, 7, 12,19,22] and references therein).
230 Pseudospectral functions
2. Preliminaries
2.1. Notations
The following notations will be used throughout the paper: H, H
denote Hilbert spaces; [H1,H2] is the set of all bounded linear operators
defined on H1 with values in H2; [H] := [H,H]; C+ (C−) is the upper
(lower) half-plane of the complex plane. If H is a subspace in H̃, then
PH(∈ [H̃]) denote the orthoprojection in H̃ onto H and PH̃,H(∈ [H̃,H])
denote the same orthoprojection considered as an operator from H̃ to H.
Recall that a linear relation T : H0 → H1 from a Hilbert space H0
to a Hilbert space H1 is a linear manifold in the Hilbert H0 ⊕ H1. If
H0 = H1 =: H one speaks of a linear relation T in H. The set of all
closed linear relations from H0 to H1 (in H) will be denoted by C̃(H0,H1)
(C̃(H)). A closed linear operator T from H0 to H1 is identified with its
graph grT ∈ C̃(H0,H1).
For a linear relation T ∈ C̃(H0,H1) we denote by domT, ranT, kerT
and mulT the domain, range, kernel and the multivalued part of T re-
spectively. Recall that mulT ia a subspace in H1 defined by
mulT := {h1 ∈ H1 : {0, h1} ∈ T}. (2.1)
Clearly, T ∈ C̃(H0,H1) is an operator if and only if mulT = {0}. For
T ∈ C̃(H0,H1) we will denote by T−1(∈ C̃(H1,H0)) and T ∗(∈ C̃(H1,H0))
the inverse and adjoint linear relations of T respectively.
Recall that an operator function Φ(·) : C+ → [H] is called a Nevan-
linna function (and referred to the class R[H]) if it is holomorphic and
ImΦ(λ) ≥ 0, λ ∈ C+.
2.2. Symmetric relations and generalized resolvents
As is known a linear relation A ∈ C̃(H) is called symmetric (self-
adjoint) if A ⊂ A∗ (resp. A = A∗). For each symmetric relation A ∈ C̃(H)
the following decompositions hold
H = H′ ⊕mulA, A = grA′ ⊕ m̂ulA, (2.2)
where m̂ulA = {0} ⊕mulA and A′ is a closed symmetric not necessarily
densely defined operator in H′ (the operator part of A). Moreover, A =
A∗ if and only if A′ = (A′)∗.
Let A = A∗ ∈ C̃(H), let B be the Borel σ-algebra of R and let E0(·) :
B → [H0] be the orthogonal spectral measure of A0. Then the spectral
measure EA(·) : B → [H] of A is defined as EA(B) = E0(B)PH′ , B ∈ B.
V. Mogilevskii 231
Definition 2.1. Let à = Ã∗ ∈ C̃(H̃) and let H be a subspace in H̃. The
relation à is called H-minimal if there is no a nontrivial subspace H′ ⊂
H̃⊖H such that E
Ã
(δ)H′ ⊂ H′ for each bounded interval δ = [α, β) ⊂ R.
Definition 2.2. The relations Tj ∈ C̃(Hj), j ∈ {1, 2}, are said to be
unitarily equivalent (by means of a unitary operator U ∈ [H1,H2]) if
T2 = ŨT1 with Ũ = U ⊕ U ∈ [H2
1,H
2
2].
Let A ∈ C̃(H) be a symmetric relation. Recall the following definitions
and results.
Definition 2.3. A relation à = Ã∗ in a Hilbert space H̃ ⊃ H satisfying
A ⊂ Ã is called an exit space self-adjoint extension of A. Moreover, such
an extension à is called minimal if it is H-minimal.
In what follows we denote by S̃elf(A) the set of all minimal exit space
self-adjoint extensions of A. Moreover, we denote by Self(A) the set of all
extensions à = Ã∗ ∈ C̃(H) of A (such an extension is called canonical). As
is known, for each A one has S̃elf(A) 6= ∅. Moreover, Self(A) 6= ∅ if and
only if A has equal deficiency indices, in which case Self(A) ⊂ S̃elf(A).
Definition 2.4. Exit space extensions Ãj = Ã∗
j ∈ C̃(H̃j), j ∈ {1, 2}, of A
are called equivalent (with respect to H) if there exists a unitary operator
V ∈ [H̃1 ⊖ H, H̃2 ⊖ H] such that Ã1 and Ã2 are unitarily equivalent by
means of U = IH ⊕ V .
Definition 2.5. The operator functions R(·) : C \ R → [H] and F (·) :
R → [H] are called a generalized resolvent and a spectral function of A
respectively if there exists an exit space self-adjoint extension à of A (in
a certain Hilbert space H̃ ⊃ H) such that
R(λ) = PH(Ã− λ)−1 ↾ H, λ ∈ C \ R (2.3)
F (t) = P
H̃,H
E
Ã
((−∞, t)) ↾ H, t ∈ R. (2.4)
Proposition 2.6. Each generalized resolvent R(λ) of A is generated
by some (minimal) extension à ∈ S̃elf(A). Moreover, the extensions
Ã1, Ã2 ∈ S̃elf(A) inducing the same generalized resolvent R(·) are equiv-
alent.
In the sequel we suppose that a generalized resolvent R(·) and a spec-
tral function F (·) are generated by an extension à ∈ S̃elf(A). Moreover,
we identify equivalent extensions. Then by Proposition 2.6 the equality
(2.3) gives a bijective correspondence between generalized resolvents R(λ)
and extensions à ∈ S̃elf(A), so that each à ∈ S̃elf(A) is uniquely defined
by the corresponding generalized resolvent (2.3) (spectral function (2.4)).
232 Pseudospectral functions
Definition 2.7. An extension à ∈ S̃elf(A) (à ∈ Self(A)) belongs to the
class S̃elf0(A) (resp. Self0(A)) if mul à = mulA.
It follows from (2.2) that the operator A′ is densely defined if and
only if mulA = mulA∗. This yields the equivalence
S̃elf(A) = S̃elf0(A) ⇐⇒ mulA = mulA∗ (2.5)
2.3. The classes R̃(H0,H1) and R̃(H)
In the following H0 is a Hilbert space, H1 is a subspace in H0, H2 :=
H0 ⊖H1, P1 := PH0,H1 and P2 = PH2 .
Definition 2.8. [24] A function τ(·) : C+ → C̃(H0,H1) is referred to
the class R̃(H0,H1) if:
(i) 2Im(h1, h0)− ||P2h0||2 ≥ 0, {h0, h1} ∈ τ(λ), λ ∈ C+;
(ii) (τ(λ) + iP1)
−1 ∈ [H1,H0], λ ∈ C+, and the operator-function
(τ(λ) + iP1)
−1 is holomorphic on C+.
According to [24] the equality
τ(λ) = {C0(λ), C1(λ)}
:= {{h0, h1} ∈ H0 ⊕H1 : C0(λ)h0 + C1(λ)h1 = 0}, λ ∈ C+ (2.6)
establishes a bijective correspondence between all functions
τ(·) ∈ R̃(H0,H1) and all pairs of holomorphic operator-functions Cj(·) :
C+ → [Hj ,H0], j ∈ {0, 1}, satisfying
2 Im(C1(λ)P1C
∗
0 (λ)) + C0(λ)P2C
∗
0 (λ) ≥ 0,
(C0(λ)− iC1(λ)P1)
−1 ∈ [H0], λ ∈ C+. (2.7)
This fact enables one to identify a function τ(·) ∈ R̃(H0,H1) and the
corresponding pair of operator-functions Cj(·) (more precisely the equiv-
alence class of such pairs [24]).
If H1 = H0 =: H, then the class R̃(H,H) coincides with the well-
known class R̃(H) of Nevanlinna C̃(H)-valued functions (Nevanlinna op-
erator pairs) τ(λ) = {C0(λ), C1(λ)}, λ ∈ C+. In this case the class
R̃0(H) is defined as the set of all τ(·) ∈ R̃(H) such that
τ(λ) ≡ θ = {C0, C1}, λ ∈ C+, (2.8)
with θ = θ∗ ∈ C̃(H) and Cj ∈ [H] satisfying Im(C1C
∗
0 ) = 0 and (C0 ±
iC1)
−1 ∈ [H].
V. Mogilevskii 233
2.4. Boundary triplets and boundary pairs
Here we recall some facts about boundary triplets and boundary pairs
following [4, 6, 7, 12,19,21,22].
Assume that A is a closed symmetric linear relation in the Hilbert
space H, Nλ(A) = ker (A∗−λ) (λ ∈ C) is a defect subspace of A, N̂λ(A) =
{{f, λf} : f ∈ Nλ(A)} and n±(A) := dimNλ(A) ≤ ∞, λ ∈ C±, are
deficiency indices of A.
Definition 2.9. A collection Π = {H0 ⊕ H1,Γ0,Γ1}, where Γj : A∗ →
Hj , j ∈ {0, 1}, are linear mappings, is called a boundary triplet for A∗,
if the mapping Γ : f̂ → {Γ0f̂ ,Γ1f̂}, f̂ ∈ A∗, from A∗ into H0 ⊕ H1 is
surjective and the following Green’s identity
(f ′, g)−(f, g′) = (Γ1f̂ ,Γ0ĝ)H0−(Γ0f̂ ,Γ1ĝ)H0+i(P2Γ0f̂ , P2Γ0ĝ)H2 (2.9)
holds for all f̂ = {f, f ′}, ĝ = {g, g′} ∈ A∗.
A boundary triplet Π = {H0 ⊕H1,Γ0,Γ1} for A∗ exists if and only if
n−(A) ≤ n+(A), in which case dimH1 = n−(A) and dimH0 = n+(A).
Proposition 2.10. Let Π = {H0 ⊕H1,Γ0,Γ1} be a boundary triplet for
A∗ and let π1 be the orthoprojection in H ⊕ H onto H ⊕ {0}. Then the
equalities
γ+(λ) = π1(Γ0 ↾ N̂λ(A))
−1, λ ∈ C+; γ−(λ) = π1(P1Γ0 ↾ N̂λ(A))
−1, λ ∈ C−
(2.10)
M+(λ)h0 = Γ1{γ+(λ)h0, λγ+(λ)h0}, h0 ∈ H0, λ ∈ C+ (2.11)
correctly define holomorphic operator functions γ+(·) : C+ → [H0,H],
γ−(·) : C− → [H1,H] (γ-fields of Π) and M+(·) : C+ → [H0,H1] (the
Weyl function of Π).
γ-field γ+(·) (γ−(·)) can be also defined as a unique [H0,H]-valued
(resp. [H1,H]-valued) operator function such that γ+(λ)H0 ⊂ Nλ(A)
(resp. γ−(λ)H1 ⊂ Nλ(A) ) and
Γ0{γ+(λ)h0, λγ+(λ)h0} = h0, h0 ∈ H0, λ ∈ C+ (2.12)
P1Γ0{γ−(λ)h1, λγ−(λ)h1} = h1, h1 ∈ H1, λ ∈ C−. (2.13)
A boundary pair for A∗ is a generalization of a boundary triplet. Namely,
a pair {H0 ⊕H1,Γ} with a linear relation Γ : H⊕H → H0 ⊕H1 is called
a boundary pair for A∗ if domΓ = A∗, the identity
(f ′, g)H − (f, g′)H = (h1, x0)H0 − (h0, x1)H0 + i(P2h0, P2x0)H0 (2.14)
234 Pseudospectral functions
holds for every {f ⊕ f ′, h0 ⊕ h1}, {g ⊕ g′, x0 ⊕ x1} ∈ Γ and a certain
maximality condition is satisfied [6, 22]. The following proposition is
immediate from [22, Section 3].
Proposition 2.11. Let {H0 ⊕ H1,Γ} be a boundary pair for A∗ with
dimH0 < ∞ and let Γ0 : H ⊕ H → H0 be the linear relations, given by
Γ0 = PH0⊕{0}Γ. Moreover, let
KΓ = mul (mul Γ) = {h1 ∈ H1 : {0⊕ 0, 0⊕ h1} ∈ Γ}, KΓ ⊂ H1.
(2.15)
Then: (1) domΓ = A∗;
(2) If KΓ = {0},then ranΓ0 ↾ N̂λ(A) = H0, λ ∈ C+; ranP1Γ0 ↾
N̂λ(A) = H1, λ ∈ C−, and the equality
grM+(λ) = {h0 ⊕ h1 : {f ⊕ λf, h0 ⊕ h1} ∈ Γ with some f ∈ Nλ(A)},
λ ∈ C+ (2.16)
defines the operator function M+(·) : C+ → [H0,H1] (the Weyl function
of the pair {H0 ⊕H1,Γ}). Moreover,
grM∗
+(λ)
= {{P1h0⊕(h1+iP2h0} : {f⊕λf, h0⊕h1} ∈ Γ with some f ∈ Nλ(A)},
λ ∈ C−. (2.17)
3. Pseudospectral and spectral functions of symmetric
systems
3.1. Notations
For an interval I = [a, b〉 ⊂ R and a finite-dimensional Hilbert space
H we denote by AC(I;H) the set of all functions f(·) : I → H, which are
absolutely continuous on each segment [α, β] ⊂ I.
Assume that ∆(·) : I → [H] is a locally integrable function such that
∆(t) ≥ 0 a.e. on I. Denote by L2
∆(I) the semi-Hilbert space of Borel
measurable functions f(·) : I → H satisfying
∫
I(∆(t)f(t), f(t))H dt < ∞
(see e.g. [10, Chapter 13.5]). The semi-definite inner product in L2
∆(I)
will be denoted (·, ·)∆. Moreover, let L2
∆(I) be the Hilbert space of equiv-
alence classes in L2
∆(I) with respect to the semi-norm in L2
∆(I), π∆ be
the quotient map from L2
∆(I) onto L2
∆(I) and π̃∆{f, g} := {π∆f, π∆g},
{f, g} ∈ (L2
∆(I))2.
For a given finite-dimensional Hilbert space K we denote by L2
∆[K,H]
the set of all Borel measurable operator-functions F (·) : I → [K,H] such
that F (t)h ∈ L2
∆(I), h ∈ K.
V. Mogilevskii 235
In the following for a distribution function σ(·) : R → [H] we denote
by L2(σ;H) the semi-Hilbert space of Borel-measurable functions g(·) :
R → H such that
∫
R
(dσ(s)g(s), g(s))(s) < ∞ and by L2(σ;H) the a
Hilbert space of all equivalence classes in L2(σ;H) with respect to the
seminorm || · ||L2(σ;H) (see e.g. [10, Chapter 13.5]). Moreover, we denote
by πσ the quotient map from L2(σ;H) onto L2(σ;H).
3.2. Symmetric systems
Let H and Ĥ be finite dimensional Hilbert spaces and let
H := H ⊕ Ĥ ⊕H (3.1)
ν = dimH, ν̂ = dim Ĥ, n = dimH = 2ν + ν̂. (3.2)
A first order symmetric system of differential equations on an interval
I = [a, b〉,−∞ < a < b ≤ ∞, (with the regular endpoint a) is of the form
Jy′ −B(t)y = λ∆(t)y, t ∈ I, λ ∈ C, (3.3)
where J is the operator (1.2) and B(·) and ∆(·) are locally integrable
[H]-valued functions on I such that B(t) = B∗(t) and ∆(t) ≥ 0 (a.e. on
I).
A function y ∈ AC(I;H) is a solution of system (3.3) if equality (3.3)
holds a.e. on I. An operator function Y (·, λ) : I → [K,H] is an operator
solution of (3.3) if y(t) = Y (t, λ)h is a solution of (3.3) for every h ∈ K
(here K is a Hilbert space with dimK <∞).
In the sequel we denote by Nλ, λ ∈ C, the linear space of all solu-
tions of the system (3.3) belonging to L2
∆(I). According to [15, 18] the
numbers N± = dimNλ, λ ∈ C±, do not depend on λ in either C+ or C−.
These numbers are called the formal deficiency indices of the system [15].
Clearly N± ≤ n.
In the following for each operator solution Y (·, λ) ∈ L2
∆[K,H] we
denote by Y (λ) the linear operator from K to Nλ given by (Y (λ)h)(t) =
Y (t, λ)h, h ∈ K.
Clearly, for any λ ∈ C the space N of all solutions y of (3.3) with
∆(t)y(t) = 0 (a.e. on I) is a subspace of Nλ; moreover, N does depend
on λ. The space N is called the null manifold of the system [15]. Denote
by θN the subspace in H given by
θN = {y(a) : y ∈ N}. (3.4)
As is known [14, 18, 30] system (3.3) gives rise to the maximal linear
relations Tmax and Tmax in L2
∆(I) and L2
∆(I) respectively. They are
236 Pseudospectral functions
given by
Tmax = {{y, f} ∈ (L2
∆(I))2 : y ∈ AC(I;H)
and Jy′(t)−B(t)y(t) = ∆(t)f(t) a.e. on I}
and Tmax = π̃∆Tmax. Moreover the Lagrange’s identity
(f, z)∆ − (y, g)∆ = [y, z]b − (Jy(a), z(a)), {y, f}, {z, g} ∈ Tmax (3.5)
holds with
[y, z]b := lim
t↑b
(Jy(t), z(t)), y, z ∈ dom Tmax.
Let Db be the set of all y ∈ dom Tmax such that [y, z]b = 0 for all z ∈
dom Tmax. The minimal relation Tmin in L2
∆(I) is defined via Tmin =
π̃∆Ta, where
Ta = {{y, f} ∈ Tmax : y ∈ Db, y(a) = 0}. (3.6)
As was shown in [14,18,22,30] Tmin is a closed symmetric linear relation
in L2
∆(I), T ∗
min = Tmax and
n+(Tmin) = N+ − dimN , n−(Tmin) = N− − dimN . (3.7)
With each subspace θ ⊂ H we associate the subspace θ× ⊂ H given by
θ× = H⊖ Jθ = {h ∈ H : (Jh, k) = 0, k ∈ θ}.
Clearly θ×× = θ. Moreover, by [22, Proposition 4.19]
θ×N = {y(a) : y ∈ Db}. (3.8)
Denote by Sym(H) the set of all subspaces θ in H satisfying θ ⊂ θ× or,
equivalently, (Jh, k) = 0, h, k ∈ θ.
The following three lemmas will be useful in the sequel.
Lemma 3.1. (1) If η ∈ Sym(H), then dim η ≤ ν and dim η× ≥ ν + ν̂.
(2) For every η ∈ Sym(H) there exists a subspace θ ⊂ H such that
θ× ∈ Sym(H), dim θ = ν + ν̂ (i.e., the dimension of θ is minimally
possible) and θ× ∩ η = {0}.
(3) Let θ be a subspace in H and θ× ∈ Sym(H). Then there exist
an operator Ũ ∈ [H] and a subspace H1 ⊂ H such that Ũ∗JŨ = J and
ŨH0 = θ, where
H0 = H ⊕ Ĥ ⊕H1. (3.9)
V. Mogilevskii 237
Proof. (1) Let Ĵ and X be operators in H given by the block represen-
tations
Ĵ = i
IH 0 0
0 I
Ĥ
0
0 0 −IH
, X = 1√
2
−iIH 0 IH
0
√
2I
Ĥ
0
iIH 0 IH
with respect to decomposition (3.1) of H. One can easily verify that
X∗ĴX = J, X∗X = XX∗ = IH. (3.10)
and the equality grVη = Xη gives a bijective correspondence between all
η ∈ Sym(H) and all isometries Vη ∈ [domVη, H] with domVη ⊂ H ⊕ Ĥ.
Hence for every η ∈ Sym(H) one has dim η = dim ranVη ≤ ν and,
consequently, dim η× ≥ ν + ν̂.
(2) Assume that η ∈ Sym(H) and let U ∈ [domU,H] be an isometry
such that domU ∈ H⊕Ĥ, −Vη ⊂ U and ranU = H. Then U = Uθ0 with
some θ0 ∈ Sym(H) and the obvious equality grVη ∩ grU = {0} yields
η ∩ θ0 = {0}. Moreover, dim θ0 = dim ranU = ν and hence θ := θ×0
possesses the required properties.
(3) Let H1 be a subspace in H with codimH1 = dim θ×, let H⊥
1 =
H⊖H1 and let H0 ⊂ H be subspace (3.9). Then H
×
0 = H⊥
1 ⊕{0}⊕{0} and
therefore H
×
0 ∈ Sym(H). Let V1 = V
H
×
0
and V2 = Vθ× . Since dimH
×
0 =
dim θ×, one has dimdomV1 = dimdomV2. Therefore there exist unitary
operators U1 ∈ [H ⊕ Ĥ] and U2 ∈ [H] such that U1domV1 = domV2
and V2U1 ↾ domV1 = U2V1. Letting Û = diag(U1, U2) one obtains the
operator Û ∈ [H] such that Û∗Ĵ Û = Ĵ and ÛgrV1 = grV2. This and
(3.10) imply that the operator Ũ := X∗ÛX satisfies Ũ∗JŨ = J and
ŨH
×
0 = θ×. Therefore ŨH0 = θ.
Remark 3.2. If H1 ⊂ H is a subspace from Lemma 3.1 (3), H⊥
1 =
H ⊖H1 and H0 is given by (3.9), then the following decompositions are
obvious:
H0 = H⊥
1 ⊕H1︸ ︷︷ ︸
H
⊕Ĥ ⊕H1, H = H⊥
1 ⊕H1︸ ︷︷ ︸
H
⊕Ĥ ⊕H1 ⊕H⊥
1︸ ︷︷ ︸
H
= H0 ⊕H⊥
1 .
(3.11)
Lemma 3.3. Let θ be a subspace in H. Then:
(1) The equalities
T = Tθ× := {π̃∆{y, f} : {y, f} ∈ Tmax, y ∈ Db, y(a) ∈ θ×} (3.12)
T ∗ = {π̃∆{y, f} : {y, f} ∈ Tmax, y(a) ∈ θ} (3.13)
238 Pseudospectral functions
defines a relation T ∈ C̃(L2
∆(I)) and its adjoint T ∗. Moreover, Tmin ⊂
T ⊂ Tmax
(2) The multivalued part mulT of T is the set of all f̃ ∈ H such that
for some (and hence for all) f(·) ∈ f̃ there exists a solution y of the
system
Jy′ −B(t)y = ∆(t)f(t), t ∈ I (3.14)
satisfying ∆(t)y(t) = 0 (a.e. on I), y(a) ∈ θ× and y ∈ Db.
(3) The relation T is symmetric if and only if θ× ∩ θ×N ∈ Sym(H).
Proof. (1) The inclusions Tmin ⊂ T ⊂ Tmax directly follow from (3.12)
and definitions of Tmin and Tmax. Next we show that the relation T ∗
adjoint to T is of the form (3.13). In view of the Lagrange’s identity (3.5)
for every {y, f} ∈ Tmax with y(a) ∈ θ one has π̃∆{y, f} ∈ T ∗. Conversely,
assume that {ỹ, f̃} ∈ T ∗ and prove the existence of {y, f} ∈ Tmax such
that y(a) ∈ θ and π̃∆{y, f} = {ỹ, f̃}. Since Tmin ⊂ T , it follows that
T ∗ ⊂ Tmax and hence there is {y1, f} ∈ Tmax such that π̃∆{y1, f} =
{ỹ, f̃}. Let h ∈ θ× ∩ θ×N . Then in view of (3.8) there exists {z, g} ∈ Tmax
such that z ∈ Db, z(a) = h and hence {z̃, g̃} := π̃∆{z, g} ∈ T . Applying
the Lagrange’s identity (3.5) to {y1, f} and {z, g} one obtains
(Jy1(a), h) = (y1, g)∆ − (f, z)∆ = (ỹ, g̃)− (f̃ , z̃) = 0, h ∈ θ× ∩ θ×N .
Therefore y1(a) ∈ (θ×∩θ×N )×. Obviously (θ×∩θ×N )× = θ+θN and hence
y1(a) = h + y2(a) with some h ∈ θ and y2 ∈ N . Let y = y1 − y2. Since
{y2, 0} ∈ Tmax, it follows that a pair {y, f} := {y1, f}−{y2, 0} belongs to
Tmax. Moreover, y(a) = y1(a) − y2(a) = h and hence y(a) ∈ θ. Finally,
π∆y2 = 0 and therefore π̃∆{y, f} = π̃∆{y1, f} = {ỹ, f̃}. This completes
the proof of (3.13).
Statement (2) directly follows from (2.1).
(3) It follows from (3.8) that T = Tθ×∩θ×N
. Therefore to prove state-
ment (3) it is sufficient to prove the following equivalent statement: if
θ× ⊂ θ×N , then the equivalence T ⊂ T ∗ ⇐⇒ θ× ⊂ θ is valid.
So assume that θ× ⊂ θ×N and let T ⊂ T ∗. If h, k ∈ θ×, then by (3.8)
there exist {y, f}, {z, g} ∈ Tmax such that y, z ∈ Db and y(a) = h, z(a) =
k. Therefore π̃∆{y, f}, π̃∆{z, g} ∈ T and hence
(f, z)∆ − (y, g)∆ = 0.
This and the Lagrange’s identity (3.5) imply that (Jh, k) = 0. Therefore
θ× ⊂ θ. If conversely θ× ⊂ θ, then the inclusion T ⊂ T ∗ directly follows
from (3.12) and (3.13).
V. Mogilevskii 239
Lemma 3.4. There exists a subspace θ ⊂ H such that θ× ∈ Sym(H),
dim θ = ν + ν̂ and the symmetric extension T = Tθ× of Tmin defined by
(3.12) satisfies mulT = mulTmin.
Proof. Let η be a subspace in H defined by
η = {y(a) : y ∈ Db, ∆(t)y(t) = 0 (a.e. on I)}. (3.15)
If h, k ∈ η, then there exist {y, f}, {z, g} ∈ Tmax such that y, z ∈ Db,
y(a) = h, z(a) = k and ∆(t)y(t) = ∆(t)z(t) = 0 (a.e. on I). Application
of the Lagrange’s identity (3.5) to such {y, f} and {z, g} gives (Jh, k) =
0, which implies that η ∈ Sym(H). Therefore by Lemma 3.1, (2) there
exists a subspace θ ⊂ H such that θ× ∈ Sym(H), dim θ = ν + ν̂ and
θ× ∩ η = {0}. Let T = Tθ× be given by (3.12) and let f̃ ∈ mulT . Then
according to Lemma 3.3, (2) there exists y ∈ Db such that y(a) ∈ θ×,
∆(t)y(t) = 0 (a.e. on I) and {y, f} ∈ Tmax for each f(·) ∈ f̃ . Since by
(3.15) y(a) ∈ θ×∩η, it follows that y(a) = 0 and hence {y, f} ∈ Ta. Hence
{π∆y, f̃} ∈ Tmin and the equality π∆y = 0 yields f̃ ∈ mulTmin. Thus
mulT ⊂ mulTmin and in view of the obvious inclusion mulTmin ⊂ mulT
one has mulT = mulTmin.
3.3. q-pseudospectral and spectral functions
In what follows we put H := L2
∆(I) and denote by Hb the set of all
f̃ ∈ H with the following property: there exists β
f̃
∈ I such that for some
(and hence for all) function f ∈ f̃ the equality ∆(t)f(t) = 0 holds a.e.
on (β
f̃
, b).
Let θ and H′
0 be subspaces in H, let K = Kθ ∈ [H′
0,H] be an operator
such that kerKθ = {0} and KθH
′
0 = θ and let ϕK(·, λ)(∈ [H′
0,H]) be an
operator solution of (3.3) satisfying ϕK(a, λ) = K, λ ∈ C. With each
f̃ ∈ Hb we associate the function f̂(·) : R → H′
0 given by
f̂(s) =
∫
I
ϕ∗
K(t, s)∆(t)f(t) dt, f(·) ∈ f̃ . (3.16)
One can easily prove that f̂(·) is a continuous function on R.
Recall that an operator V ∈ [H1,H2] is a partial isometry if ||V f || =
||f || for all f ∈ H1 ⊖ kerV .
Definition 3.5. A distribution function σ(·) : R → [H′
0] is called a q-
pseudospectral function of the system (3.3) (with respect to the operator
K = Kθ) if f̂ ∈ L2(σ;H′
0) for all f̃ ∈ Hb and the operator V f̃ := πσf̂ , f̃ ∈
Hb, admits a continuation to a partial isometry Vσ ∈ [H, L2(σ;H′
0)]. The
operator Vσ is called the (generalized) Fourier transform corresponding
to σ(·).
240 Pseudospectral functions
Clearly, if σ(·) is a q-pseudospectral function, then for each f(·) ∈
L2
∆(I) there exists a unique (up to the seminorm in L2(σ;H′
0)) function
f̂(·) ∈ L2(σ;H′
0) such that
lim
β↑b
∣∣∣
∣∣∣f̂(·)−
∫
[a,β)
ϕ∗
K(t, ·)∆(t)f(t) dt
∣∣∣
∣∣∣
L2(σ;H′
0)
= 0.
The function f̂(·) is called the Fourier transform of the function f(·).
Remark 3.6. Similarly to [10, 33] (see also [26, Proposition 3.4]) one
proves that for each q-pseudospectral function σ(·)
V ∗
σ g̃ = π∆
(∫
R
ϕK(·, s) dσ(s)g(s)
)
, g̃ ∈ L2(σ;H′
0), g(·) ∈ g̃, (3.17)
where the integral converges in the seminorm of L2
∆(I). Hence for each
function f(·) ∈ L2
∆(I) with π∆f ∈ H ⊖ kerVσ the equality (the inverse
Fourier transform)
f(t) =
∫
R
ϕK(t, s) dσ(s)f̂(s)
is valid. Therefore the natural problem is a characterization of q-pseudo-
spectral functions σ(·) with the minimally possible kernel of Vσ.
The following lemma can be proved in the same way as Lemma 3.7
in [27].
Lemma 3.7. Assume that θ and H′
0 are subspaces in H, σ(·) is a q-
pseudospectral function (with respect to Kθ ∈ [H′
0,H]), Vσ is the cor-
responding Fourier transform and T ∈ C̃(H) is given by (3.12). Then
there exist a Hilbert space H̃ ⊃ H and a self-adjoint operator T̃0 in
H̃0 := H̃ ⊖ kerVσ such that T̃0 ⊂ T ∗
H̃
(here T ∗
H̃
∈ C̃(H̃) is the linear
relation adjoint to T in H̃).
By using Lemma 3.7 one can prove similarly to [27, Proposition 3.8]
the following theorem.
Theorem 3.8. Let the assumptions of Lemma 3.7 be satisfied and let
mulT be the multivalued part of T (see Lemma 3.3, (2)). Then
mulT ⊂ kerVσ (3.18)
Definition 3.9. Under the assumptions of Theorem 3.8 a q-pseudospec-
tral function σ(·) of the system (3.3) with kerVσ = mulT is called a
pseudospectral function .
V. Mogilevskii 241
Definition 3.10. Let θ and H′
0 be subspaces in H. A distribution func-
tion σ(·) : R → [H′
0] is called a spectral function of the system (3.3) (with
respect to Kθ ∈ [H′
0,H]) if for every f̃ ∈ Hb the inclusion f̂ ∈ L2(σ;H′
0)
holds and the Parseval equality ||f̂ ||L2(σ;H′
0)
= ||f̃ ||H is valid (for f̂ see
(3.16)).
The number nσ := dimH′
0(= dim θ) is called a dimension of the
spectral function σ(·).
If for a given Kθ ∈ [H′
0,H] a pseudospectral function σ(·) exists, then
in view of (3.18) it is a q-pseudospectral function with the minimally
possible kerVσ (see the problem posted in Remark 3.6). Moreover, (3.18)
yields the following proposition.
Proposition 3.11. Let θ and H′
0 be subspaces in H and let T ∈ C̃(H)
be given by (3.12). If mulT 6= {0}, then the set of spectral functions
(with respect to Kθ ∈ [H′
0,H]) is empty. If mulT = {0}, then the sets of
spectral and pseudospectral functions (with respect to Kθ) coincide.
A connection between different q-pseudospectral functions correspon-
ding to the same subspace θ ⊂ H is specified in the following proposition.
Proposition 3.12. Assume that θ and H′
0j are subspaces in H and Kj =
Kjθ ∈ [H′
0j ,H] are operators such that kerKj = {0} and KjH
′
0j = θ, j ∈
{1, 2}. Then: (1) there exists a unique isomorphism X ∈ [H′
01,H
′
02] such
that K1 = K2X; (2) the equality σ2(s) = Xσ1(s)X
∗ gives a bijective
correspondence between all q-pseudospectral functions σ1(·) (with respect
to K1) and σ2(·) (with respect to K2) of the system (3.3). Moreover σ2(·)
is a pseudospectral or spectral function if and only if so is σ1(·).
Proof. Statement (1) is obvious To prove statement (2) assume that σ1(·)
is a q-pseudospectral function (with respect to K1) and σ2(·) is an [H′
02]
-valued distribution function given by σ2(s) = Xσ1(s)X
∗. One can easily
verify that the equality (Ug)(s) = X−1∗g(s), g = g(·) ∈ L2(σ1;H
′
01), de-
fines a surjective linear operator U : L2(σ1;H
′
01) → L2(σ2;H
′
02) satisfying
||Ug||L2(σ2;H′
02)
= ||g||L2(σ1;H′
01)
. Therefore the equality Ug̃ = πσ2Ug, g̃ ∈
L2(σ1;H
′
01), g ∈ g̃, defines a unitary operator U ∈ [L2(σ1;H
′
01), L
2(σ2;H
′
02)].
Next assume that f̃ ∈ Hb, f(·) ∈ f̃ and f̂j(·) is the Fourier transform
of f(·) given by (3.16) with ϕK(·, λ) = ϕKj
(·, λ), j ∈ {1, 2}. Since obvi-
ously ϕK1(t, s) = ϕK2(t, s)X, it follows that f̂2(s) = X−1∗f̂1(s). Hence
f̂2 = U f̂1 ∈ L2(σ2;H
′
02) and πσ2 f̂2 = Uπσ1 f̂1 = UVσ1 f̃ . This implies that
the operator V2f̃ := πσ2 f̂2, f̃ ∈ Hb, admits a continuation to the partial
isometry Vσ2 = UVσ1(∈ [H, L2(σ2;H
′
02)]) with kerVσ2 = kerVσ1 . There-
fore σ2(·) is a q-pseudospectral function (with respect to K2); moreover,
σ2(·) is a pseudospectral or spectral function if and only if so is σ1(·).
242 Pseudospectral functions
Remark 3.13. It follows from Proposition 3.12 that a q-pseudospectral
(in particular pseudospectral) function σ(·) with respect to the operator
Kθ ∈ [H′
0,H] is uniquely characterized by the subspace θ ⊂ H.
Under the assumptions of Theorem 3.8 we let H0 := H ⊖ mulT , so
that
H = mulT ⊕ H0.
Moreover, for a pseudospectral function σ(·) we denote by V0 = V0,σ the
isometry from H0 to L2(σ;H′
0) given by V0,σ := Vσ ↾ H0. Next assume
that H̃ ⊃ H is a Hilbert space and T̃ = T̃ ∗ ∈ C̃(H̃) with mul T̃ = mulT .
In the following we put H̃0 := H̃⊖mulT , so that H0 ⊂ H̃0 and
H̃ = mulT ⊕ H̃0.
Denote also by T̃0 the operator part of T̃ . Clearly, T̃0 is a self-adjoint
operator in H̃0.
Proposition 3.14. Assume that θ and H′
0 are subspaces in H, σ(·) is
a pseudospectral function (with respect to Kθ ∈ [H′
0,H]) and T ∈ C̃(H)
is given by (3.12). Moreover, let L0 = VσH and let Λσ = Λ∗
σ be the
multiplication operator in L2(σ;H′
0) defined by
domΛσ = {f̃ ∈ L2(σ;H′
0) : sf(s) ∈ L2(σ;H′
0)
for some (and hence for all) f(·) ∈ f̃}
Λσf̃ = πσ(sf(s)), f̃ ∈ domΛσ, f(·) ∈ f̃ .
Then T is a symmetric extension of Tmin and there exist a Hilbert space
H̃ ⊃ H and an exit space self-adjoint extension T̃ ∈ C̃(H̃) of T such that
mul T̃ = mulT and the relative spectral function
F (t) = P
H̃,H
E
T̃
((−∞, t)) ↾ H
of T satisfies
((F (β)−F (α))f̃ , f̃) =
∫
[α,β)
(dσ(s)f̂(s), f̂(s)), f̃ ∈ Hb,−∞ < α < β <∞.
(3.19)
Moreover, there exists a unitary operator Ṽ ∈ [H̃0, L
2(σ;H′
0)] such that
Ṽ ↾ H0 = V0,σ and the operators T̃0 and Λσ are unitarily equivalent by
means of Ṽ .
If in addition the operator Λσ is L0-minimal, then the extension T̃ is
unique(up to the equivalence) and T̃ ∈ S̃elf0(T ) (that is, T̃ is H-minimal).
V. Mogilevskii 243
Proof. By using Lemma 3.7 one proves as in [27, Proposition 5.6] the
following statement:
(S) There is a Hilbert space H̃ ⊃ H and a relation T̃ = T̃ ∗ ∈
C̃(H̃) such that mul T̃ = mulT , T ⊂ T̃ and (3.19) holds with F (t) =
P
H̃,H
E
T̃
((−∞, t)) ↾ H.
Moreover, by Lemma 3.3, (1) Tmin ⊂ T . Therefore T is a symmetric
extension Tmin and F (·) is a spectral function of T . Other statements of
the proposition can be proved as in [27, Proposition 5.6].
Definition 3.15. [3, 11] System (3.3) is called definite if N = {0} or,
equivalently, if for some (and hence for all) λ ∈ C there exists only a
trivial solution y = 0 of this system satisfying ∆(t)y(t) = 0 (a.e. on I).
Proposition 3.16. Let θ be a subspaces in H and let σ(·) be a pseu-
dospectral function (with respect to Kθ ∈ [H′
0,H]). Then θ× ∩ θ×N ∈
Sym(H). If in addition the system is definite, then θ× ∈ Sym(H).
Proof. The first statement is immediate from Proposition 3.14 and Lem-
ma 3.3, (3). For a definite system θN = {0} and hence θ×N = H. This
yields the second statement.
Remark 3.17. Proposition 3.16 shows that a necessary condition for
existence of a pseudospectral function for a given θ is θ×∩θ×N ∈ Sym(H).
Clearly this condition is satisfied if θ× ∈ Sym(H).
4. m-functions of symmetric systems
4.1. Boundary pairs and boundary triplets for symmetric
systems
Definition 4.1. Let θ be a subspace in H. System (3.3) will be called
θ-definite if the conditions y ∈ N and y(a) ∈ θ yield y = 0.
Remark 4.2. If system is definite then obviously it is θ-definite for any
θ ∈ H. Hence θ-definiteness is generally speaking a weaker condition
then definiteness. At the same time in the case θ = H(⇔ θ× = {0})
definiteness of the system is the same as θ-definiteness.
The following assertion directly follows from definition of Tmin and
(3.13), (2.1).
Assertion 4.3. (1) The equality mulTmin = {0} is equivalent to the
following condition:
244 Pseudospectral functions
(C0) If f(·) ∈ L2
∆(I) and there exists a solution y(·) of (3.14) such
that ∆(t)y(t) = 0 (a.e. on I), y(a) = 0 and y ∈ Db, then ∆(t)f(t) = 0
(a.e. on I).
(2) Let θ× ∈ Sym(H), let system (3.3) be θ-definite and let T be the
relation (3.12). Then the equalities mulT = {0}, mulT = mulT ∗ and
mulT ∗ = {0} are equivalent to the following conditions (C1), (C2) and
(C3) respectively:
(C1) If f(·) ∈ L2
∆(I) and there exists a solution y(·) of the system
(3.14) such that ∆(t)y(t) = 0 (a.e. on I), y(a) ∈ θ× and y ∈ Db, then
∆(t)f(t) = 0 (a.e. on I).
(C2) If f(·) ∈ L2
∆(I) and y(·) is a solution of (3.14) such that y(a) ∈ θ
and ∆(t)y(t) = 0 (a.e. on I), then y(·) ∈ Db and y(a) ∈ θ×.
(C3) If f(·) ∈ L2
∆(I) and there exists a solution y(·) of (3.14) satis-
fying ∆(t)y(t) = 0 (a.e. on I) and y(a) ∈ θ, then ∆(t)f(t) = 0 (a.e. on
I).
The following proposition can be proved in the same way as Proposi-
tion 5.5 in [27].
Proposition 4.4. Assume that θ and H′
0 are subspaces in H, σ(·) is a q-
pseudospectral function (with respect to Kθ ∈ [H′
0,H])and L0 := VσH. If
system is θ-definite, then the multiplication operator Λσ is L0-minimal.
Below within this section we suppose the following assumptions:
(A1) θ is a subspace in H and θ× ∈ Sym(H). Moreover, system (3.3)
is θ-definite and satisfies N− ≤ N+.
(A2) H1 is a subspace in H, H0 ⊂ H is the subspace (3.9), Ũ ∈ [H]
is an operator satisfying Ũ∗JŨ = J and ŨH0 = θ, Γa : dom Tmax → H is
the linear operator given by Γay = Ũ−1y(a), y ∈ dom Tmax, and
Γa = (Γ1
0a, Γ
2
0a, Γ̂a, Γ
2
1a, Γ
1
1a)
⊤ : dom Tmax → H⊥
1 ⊕H1 ⊕ Ĥ ⊕H1 ⊕H⊥
1
(4.1)
is the block representation of Γa in accordance with the decomposition
(3.11) of H.
(A3) H̃b and Hb ⊂ H̃b are finite dimensional Hilbert spaces and
Γb = (Γ0b, Γ̂b, Γ1b)
⊤ : dom Tmax → H̃b ⊕ Ĥ ⊕Hb (4.2)
is a surjective linear operator satisfying for all y, z ∈ dom Tmax the fol-
lowing identity
[y, z]b = (Γ0by,Γ1bz)− (Γ1by,Γ0bz) + i(PH⊥
b
Γ0by, PH⊥
b
Γ0bz) + i(Γ̂by, Γ̂bz)
(4.3)
(here H⊥
b = H̃b ⊖Hb).
V. Mogilevskii 245
Remark 4.5. Existence of the operators Ũ in assumption (A2) and Γb
in assumption (A3) follows from Lemma 3.1, (3) and [1, Lemma 3.4]
respectively. Moreover, in the case N+ = N− (and only in this case) one
has H̃b = Hb and the identity (4.3) takes the form
[y, z]b = (Γ0by,Γ1bz)− (Γ1by,Γ0bz) + i(Γ̂by, Γ̂bz), y, z ∈ dom Tmax.
Observe also that Γby is a singular boundary value of a function y ∈
dom Tmax at the endpoint b (for more details see [1, Remark 3.5]).
The following lemma directly follows from definition of the operator
Γa and its block representation (4.1).
Lemma 4.6. Let Y (·, λ) ∈ L2
∆[K,H] be an operator solution of (3.3).
Then
Ũ−1Y (a, λ) = ΓaY (λ) = (PH,H0ΓaY (λ),Γ1
1aY (λ))⊤ : K → H0 ⊕H⊥
1 ,
(4.4)
where
PH,H0Γa = (Γ1
0a,Γ
2
0a, Γ̂a,Γ
2
1a)
⊤ : dom Tmax → H⊥
1 ⊕H1 ⊕ Ĥ ⊕H1.
(4.5)
Proposition 4.7. Let H0 and H1 ⊂ H0 be finite dimensional Hilbert
spaces and let Γ′
j : dom Tmax → Hj , j ∈ {0, 1}, be linear operators given
by
H0 = H⊥
1 ⊕H1 ⊕ Ĥ ⊕ H̃b, H1 = H⊥
1 ⊕H1 ⊕ Ĥ ⊕Hb (4.6)
Γ′
0 = (−Γ1
1a, −Γ2
1a, i(Γ̂a − Γ̂b), Γ0b)
⊤ : dom Tmax → H⊥
1 ⊕H1 ⊕ Ĥ ⊕ H̃b
(4.7)
Γ′
1 = (Γ1
0a, Γ
2
0a,
1
2(Γ̂a + Γ̂b), −Γ1b)
⊤ : dom Tmax → H⊥
1 ⊕H1 ⊕ Ĥ ⊕Hb.
(4.8)
Then dimH0 = N+, dimH1 = N− and a pair {H0⊕H1,Γ} with a linear
relation Γ : H⊕ H → H0 ⊕H1 defined by
Γ = {{π̃∆{y, f},Γ′
0y ⊕ Γ′
1y} : {y, f} ∈ Tmax} (4.9)
is a boundary pair for Tmax such that KΓ = {0} (for KΓ see (2.15)).
Proof. The fact that {H0 ⊕ H1,Γ} is a boundary pair for Tmax as well
as the equalities dimH0 = N+, dimH1 = N− directly follow from [22,
Theorem 5.3]. Next, according to [22] mul Γ = {{Γ′
0y,Γ
′
1y} : y ∈ N}
246 Pseudospectral functions
and hence KΓ = {Γ′
1y : y ∈ N and Γ′
0y = 0}. Moreover, the equalities
Ũ−1θ = H0 and (3.11), (4.1) yield the equivalence
y(a) ∈ θ ⇐⇒ Γ1
1ay = 0, y ∈ dom Tmax. (4.10)
Since the system is θ-definite, this implies the equality KΓ = {0}.
Definition 4.8. The boundary pair {H0⊕H1,Γ} constructed in Propo-
sition 4.7 is called a decomposing boundary pair for Tmax.
Let Ḣ0 and Ḣ1 ⊂ Ḣ0 be finite dimensional Hilbert spaces and Γ̇′
j :
dom Tmax → Ḣj , j ∈ {0, 1}, be linear operators given by
Ḣ0 = H1 ⊕ Ĥ ⊕ H̃b, Ḣ1 = H1 ⊕ Ĥ ⊕Hb (4.11)
Γ̇′
0 = (−Γ2
1a, i(Γ̂a − Γ̂b), Γ0b)
⊤ : dom Tmax → H1 ⊕ Ĥ ⊕ H̃b (4.12)
Γ̇′
1 = (Γ2
0a,
1
2(Γ̂a + Γ̂b), −Γ1b)
⊤ : dom Tmax → H1 ⊕ Ĥ ⊕Hb. (4.13)
It follows from (4.6)–(4.8) that
H0 = H⊥
1 ⊕ Ḣ0, H1 = H⊥
1 ⊕ Ḣ1 (4.14)
Γ′
0 = (−Γ1
1a, Γ̇
′
0)
⊤ : dom Tmax → H⊥
1 ⊕ Ḣ0 (4.15)
Γ′
1 = (Γ1
0a, Γ̇
′
1)
⊤ : dom Tmax → H⊥
1 ⊕ Ḣ1. (4.16)
Proposition 4.9. Let T ∈ C̃(H) be given by (3.12). Then:
(1) T is a symmetric extension of Tmin and the following equalities
hold:
T = {π̃∆{y, f} : {y, f} ∈ Tmax, y ∈ Db, Γ
1
1ay = 0,
Γ2
0ay = Γ2
1ay = 0, Γ̂ay = 0} (4.17)
T ∗ = {π̃∆{y, f} : {y, f} ∈ Tmax, Γ
1
1ay = 0} (4.18)
(2) For every {ỹ, f̃} ∈ T ∗ there exists a unique y ∈ dom Tmax such
that Γ1
1ay = 0, π∆y = ỹ and {y, f} ∈ Tmax for any f ∈ f̃ .
(3) The collection Π̇ = {Ḣ0⊕Ḣ1, Γ̇0, Γ̇1} with operators Γ̇j : T
∗ → Ḣj
of the form
Γ̇0{ỹ, f̃} = Γ̇′
0y, Γ̇1{ỹ, f̃} = Γ̇′
1y, {ỹ, f̃} ∈ T ∗ (4.19)
is a boundary triplet for T ∗. In (4.19) y ∈ dom Tmax is uniquely defined
by {ỹ, f̃} in accordance with statement (2).
V. Mogilevskii 247
Proof. (1) Since Ũ−1θ = H0 and Ũ−1θ× = H
×
0 = H⊥
1 ⊕ {0} ⊕ {0} ⊕ {0},
the equivalences (4.10) and
y(a) ∈ θ× ⇐⇒ (Γ1
1ay = 0, Γ2
0ay = Γ2
1ay = 0, Γ̂ay = 0), y ∈ dom Tmax
are valid. This and (3.12), (3.13) yield (4.17) and (4.18).
By using θ-definiteness of the system one proves statement (2) simi-
larly to [27, Proposition 4.5, (2)].
(3) Equalities (4.15), (4.16) and identity (2.14) for the decomposing
boundary pair yield the Green’s identity (2.9) for operators Γ̇0 and Γ̇1.
To prove surjectivity of the operator (Γ̇0, Γ̇1)
⊤ it is sufficient to show that
ker Γ̇0 ∩ ker Γ̇1 = T, dim Ḣ0 = n+(T ), dim Ḣ1 = n−(T ). (4.20)
Clearly, {ỹ, f̃} ∈ ker Γ̇0 ∩ ker Γ̇1 if and only if there is {y, f} ∈ Tmax
such that π̃∆{y, f} = {ỹ, f̃} and Γ1
1ay = 0, Γ2
0ay = Γ2
1ay = 0, Γ̂ay = 0,
Γby = 0. Moreover, in view of (4.3) and surjectivity of the operator Γb
the equivalence Γby = 0 ⇐⇒ y ∈ Db is valid. This yields the first
equality in (4.20). Next assume that
T = {{y, f} ∈ Tmax : y ∈ Db, y(a) ∈ θ× ∩ θ×N }.
It follows from (3.8) and (3.6) that dim(dom T /dom Ta) = dim(θ× ∩ θ×N )
and T = π̃∆T . If {y, f} ∈ T and π̃∆{y, f} = 0, then y ∈ N and
y(a) ∈ θ× ⊂ θ. Therefore in view of θ-definiteness y = 0 and conse-
quently ker π̃∆ ↾ T = {0}. This and the obvious equality dim(T /Ta) =
dim(dom T /dom Ta) imply that
dim(T/Tmin) = dim(θ× ∩ θ×N ). (4.21)
In view of θ-definiteness one has θ∩θN = {0}. Since obviously θ×∩θ×N =
(θ ∔ θN )×, it follows that
dim(θ× ∩ θ×N ) = n− dim θ − dim θN = codim θ − dimN . (4.22)
Combining (4.21) and (4.22) with the well known equality n±(T ) =
n±(Tmin)− dim(T/Tmin) and taking (3.7) into account on gets n±(T ) =
N± − codim θ. Moreover, the equality Ũ−1θ = H0 yields codim θ =
dimH⊥
1 and according to Proposition 4.7 dimH0 = N+, dimH1 = N−.
This implies that
n+(T ) = dimH0 − dimH⊥
1 , n−(T ) = dimH1 − dimH⊥
1 (4.23)
Now combining (4.23) with (4.14) one obtains the second and third equal-
ities in (4.20).
248 Pseudospectral functions
4.2. L2
∆-solutions of boundary problems
Definition 4.10. Let Ḣ0 and Ḣ1 be given by (4.11). A boundary pa-
rameter is a pair
τ = τ(λ) = {C0(λ), C1(λ)} ∈ R̃(Ḣ0, Ḣ1), λ ∈ C+, (4.24)
where Cj(λ)(∈ [Ḣj , Ḣ0]), j ∈ {0, 1}, are holomorphic operator functions
satisfying (2.7).
In the case N+ = N− (and only in this case) H̃b = Hb, Ḣ0 = Ḣ1 =: Ḣ
and τ ∈ R̃(Ḣ). If in addition τ = τ(λ) ∈ R̃0(Ḣ) is an operator pair (2.8),
then a boundary parameter τ will be called self-adjoint.
Let τ be a boundary parameter (4.24). For a given f ∈ L2
∆(I) consider
the boundary value problem
Jy′ −B(t)y = λ∆(t)y +∆(t)f(t), t ∈ I (4.25)
Γ1
1ay = 0, C0(λ)Γ̇
′
0y − C1(λ)Γ̇
′
1y = 0, λ ∈ C+ (4.26)
A function y(·, ·) : I × C+ → H is called a solution of this problem if
for each λ ∈ C+ the function y(·, λ) belongs to AC(I;H) ∩ L2
∆(I) and
satisfies the equation (4.25) a.e. on I (so that y ∈ dom Tmax) and the
boundary conditions (4.26).
The following theorem is a consequence of Theorem 3.11 in [24] ap-
plied to the boundary triplet Π̇ for T ∗.
Theorem 4.11. Let under the assumptions (A1)–(A3) T be a symmetric
relation (3.12) (or equivalently (4.17)). If τ is a boundary parameter
(4.24), then for every f ∈ L2
∆(I) the problem (4.25), (4.26) has a unique
solution y(t, λ) = yf (t, λ) and the equality
R(λ)f̃ = π∆(yf (·, λ)), f̃ ∈ H, f ∈ f̃ , λ ∈ C+
defines a generalized resolvent R(λ) =: Rτ (λ) of T . Conversely, for each
generalized resolvent R(λ) of T there exists a unique boundary parameter
τ such that R(λ) = Rτ (λ). Moreover, if N+ = N−, then Rτ (λ) is a
canonical resolvent if and only if τ is a self-adjoint boundary parameter
(2.8). In this case Rτ (λ) = (T̃τ − λ)−1, where T̃τ ∈ Self(T ) is given by
T̃τ = {π̃∆{y, f} : {y, f} ∈ Tmax,Γ
1
1ay = 0, C0Γ̇
′
0y − C1Γ̇
′
1y = 0}. (4.27)
Proposition 4.12. For any λ ∈ C \ R there exists a unique collec-
tion of operator solutions ξ1(·, λ) ∈ L2
∆[H
⊥
1 ,H], ξ2(·, λ) ∈ L2
∆[H1,H] and
V. Mogilevskii 249
ξ3(·, λ) ∈ L2
∆[Ĥ,H] of the system (3.3) satisfying the boundary conditions
Γ1
1aξ1(λ) = −IH⊥
1
, Γ2
1aξ1(λ) = 0, Γ̂aξ1(λ) = Γ̂bξ1(λ), λ ∈ C \ R
(4.28)
Γ0bξ1(λ) = 0, λ ∈ C+; PH̃b,Hb
Γ0bξ1(λ) = 0, λ ∈ C−. (4.29)
Γ1
1aξ2(λ) = 0, Γ2
1aξ2(λ) = −IH1 , Γ̂aξ2(λ) = Γ̂bξ2(λ), λ ∈ C \ R
(4.30)
Γ0bξ2(λ) = 0, λ ∈ C+; PH̃b,Hb
Γ0bξ2(λ) = 0, λ ∈ C−. (4.31)
Γ1
1aξ3(λ) = 0, Γ2
1aξ3(λ) = 0, i(Γ̂a − Γ̂b)ξ3(λ) = I
Ĥ
, λ ∈ C \ R
(4.32)
Γ0bξ3(λ) = 0, λ ∈ C+; PH̃b,Hb
Γ0bξ3(λ) = 0, λ ∈ C−. (4.33)
Moreover, for any λ ∈ C+ (λ ∈ C−) there exists a unique operator
solution u+(·, λ) ∈ L2
∆[H̃b,H] (resp. u−(·, λ) ∈ L2
∆[Hb,H]) satisfying the
boundary conditions
Γ1
1au±(λ) = 0, Γ2
1au±(λ) = 0, Γ̂au±(λ) = Γ̂bu±(λ), λ ∈ C±
(4.34)
Γ0bu+(λ) = IH̃b
, λ ∈ C+; PH̃b,Hb
Γ0bu−(λ) = IHb
, λ ∈ C−. (4.35)
Proof. Let {H0 ⊕ H1,Γ} be the decomposing boundary pair (4.9) for
Tmax. Then the linear relation Γ0 = PH0⊕{0}Γ : H2 → H0 for this triplet
is
Γ0 = {{π̃∆{y, f},Γ′
0y} : {y, f} ∈ Tmax}. (4.36)
By using (4.36) one proves in the same way as in [27, Proposition 4.8]
that
Γ0 ↾ N̂λ(Tmin) = {{π̃∆{y, λy},Γ′
0y} : y ∈ Nλ}, λ ∈ C \ R. (4.37)
Since by Proposition 4.7 KΓ = {0}, it follows from Proposition 2.11 that
ranΓ0 ↾ N̂λ(Tmin) = H0 and (4.37) yields Γ′
0Nλ = H0, λ ∈ C+. More-
over, by Proposition 4.6 dimNλ = dimH0 and hence for each λ ∈ C+
the operator Γ′
0 ↾ Nλ isomorphically maps Nλ onto H0. Similarly by
using (4.37) one proves that for each λ ∈ C− the operator P1Γ
′
0 ↾ Nλ
isomorphically maps Nλ onto H1. Therefore the equalities Z+(λ) =
(Γ′
0 ↾ Nλ)
−1, λ ∈ C+, and Z−(λ) = (P1Γ
′
0 ↾ Nλ)
−1, λ ∈ C−, define the
isomorphisms Z+(λ) : H0 → Nλ and Z−(λ) : H1 → Nλ such that
Γ′
0Z+(λ) = IH0 , λ ∈ C+; P1Γ
′
0Z−(λ) = IH1 , λ ∈ C−. (4.38)
250 Pseudospectral functions
Assume that the block representations of Z±(λ) are
Z+(λ) = (ξ1(λ), ξ2(λ), ξ3(λ), u+(λ)) : H
⊥
1 ⊕H1 ⊕ Ĥ ⊕ H̃b → Nλ,
λ ∈ C+ (4.39)
Z−(λ) = (ξ1(λ), ξ2(λ), ξ3(λ), u−(λ)) : H⊥
1 ⊕H1 ⊕ Ĥ ⊕Hb → Nλ,
λ ∈ C− (4.40)
and let ξ1(·, λ) ∈ L2
∆[H
⊥
1 ,H], ξ2(·, λ) ∈ L2
∆[H1,H] , ξ3(·, λ) ∈ L2
∆[Ĥ,H],
u+(·, λ) ∈ L2
∆[H̃b,H] and u−(·, λ) ∈ L2
∆[Hb,H] be the respective operator
solutions of (3.3). It follows from (4.7) that
P1Γ
′
0 = (−Γ1
1a, −Γ2
1a, i(Γ̂a − Γ̂b), PH̃b,Hb
Γ0b)
⊤ : dom Tmax
→ H⊥
1 ⊕H1 ⊕ Ĥ ⊕Hb (4.41)
Now combining (4.38) with (4.7), (4.39), (4.41), (4.40) and taking the
block representations of IH0 and IH1 into account one gets the equalities
(4.28)–(4.35). Finally, uniqueness of specified operator solutions is im-
plied by the equalities ker Γ′
0 ↾ Nλ = {0}, λ ∈ C+, and kerP1Γ
′
0 ↾ Nλ =
{0}, λ ∈ C−.
Proposition 4.13. The Weyl function M+ = M+(λ), λ ∈ C+, of the
decomposing boundary pair {H0 ⊕H1,Γ} for Tmax admits the block rep-
resentation
M+ =
M11 M12 M13 M14
M21 M22 M23 M24
M31 M32 M33 M34
M41 M42 M43 M44
: H⊥
1 ⊕H1 ⊕ Ĥ ⊕ H̃b︸ ︷︷ ︸
H0
→ H⊥
1 ⊕H1 ⊕ Ĥ ⊕Hb︸ ︷︷ ︸
H1
, (4.42)
with entries Mjk =Mjk(λ), λ ∈ C+, defined by
Mjk(λ) = Γj
0aξk(λ), j ∈ {1, 2}, k ∈ {1, 2, 3};
Mj4(λ) = Γj
0au+(λ), j ∈ {1, 2} (4.43)
M3k(λ) = Γ̂aξk(λ), k ∈ {1, 2}; M33(λ) = Γ̂aξ3(λ) +
i
2IĤ ,
M34(λ) = Γ̂au+(λ) (4.44)
M4k(λ) = −Γ1bξk(λ), k ∈ {1, 2, 3}; M44(λ) = −Γ1bu+(λ). (4.45)
V. Mogilevskii 251
Moreover, for every λ ∈ C− one has
M∗
jk(λ) = Γk
0aξj(λ), k ∈ {1, 2}, j ∈ {1, 2, 3};
M∗
4k(λ) = Γk
0au−(λ), k ∈ {1, 2}, (4.46)
M∗
j3(λ) = Γ̂aξj(λ), j ∈ {1, 2}; M∗
33(λ) = Γ̂aξ3(λ) +
i
2IĤ ,
M∗
43(λ) = Γ̂au−(λ). (4.47)
Proof. Let Z±(λ) be the same as in the proof of Proposition 4.12. Then
by (4.9)
{π∆Z+(λ)h0 ⊕ λπ∆Z+(λ)h0,Γ
′
0Z+(λ)h0 ⊕ Γ′
1Z+(λ)h0} ∈ Γ,
h0 ∈ H0, λ ∈ C+,
{π∆Z−(λ)h1 ⊕ λπ∆Z−(λ)h1,Γ′
0Z−(λ)h1 ⊕ Γ′
1Z−(λ)h1} ∈ Γ,
h1 ∈ H1, λ ∈ C−
and in view of (2.16) and (2.17) one has
Γ′
1Z+(λ) =M+(λ)Γ
′
0Z+(λ); (Γ′
1 + iP2Γ
′
0)Z−(λ) =M∗
+(λ)P1Γ
′
0Z−(λ).
(the first equality holds for λ ∈ C+, while the second one for λ ∈ C−).
This and (4.38) imply that
Γ′
1Z+(λ) =M+(λ), λ ∈ C+; (Γ′
1+iP2Γ
′
0)Z−(λ) =M∗
+(λ), λ ∈ C−.
(4.48)
It follows from (4.7) and (4.8) that
Γ′
1 + iP2Γ
′
0 = (Γ1
0a, Γ
2
0a,
1
2(Γ̂a + Γ̂b), ∗)⊤ : dom Tmax
→ H⊥
1 ⊕H1 ⊕ Ĥ ⊕ H̃b (4.49)
(the entry ∗ does not matter). Assume that (4.42) is the block represen-
tation ofM+(λ). Combining the first equality in (4.48) with (4.8), (4.39)
and taking the last equalities in (4.28), (4.30), (4.32) and (4.34) into ac-
count one gets (4.43)–(4.45). Similarly combining the second equality in
(4.48) with (4.49) and (4.40) one obtains (4.46) and (4.47).
Using the entries Mij = Mij(λ) from the block representation (4.42)
of M+(λ) introduce the holomorphic operator-functions m0 = m0(λ)(∈
[H0]), S1 = S1(λ)(∈ [Ḣ0,H0]), S2 = S2(λ)(∈ [H0, Ḣ1]) and Ṁ+ =
252 Pseudospectral functions
M+(λ)(∈ [Ḣ0, Ḣ1]), λ ∈ C+, by setting
m0 =
M11 M12 M13 0
M21 M22 M23 −1
2IH1
M31 M32 M33 0
0 −1
2IH1 0 0
: H⊥
1 ⊕H1 ⊕ Ĥ ⊕H1︸ ︷︷ ︸
H0
→ H⊥
1 ⊕H1 ⊕ Ĥ ⊕H1︸ ︷︷ ︸
H0
(4.50)
S1 =
M12 M13 M14
M22 M23 M24
M32 M33 − i
2IĤ M34
−IH1 0 0
: H1 ⊕ Ĥ ⊕ H̃b︸ ︷︷ ︸
Ḣ0
→ H⊥
1 ⊕H1 ⊕ Ĥ ⊕H1︸ ︷︷ ︸
H0
(4.51)
S2 =
M21 M22 M23 −IH1
M31 M32 M33 +
i
2IĤ 0
M41 M42 M43 0
: H⊥
1 ⊕H1 ⊕ Ĥ ⊕H1︸ ︷︷ ︸
H0
→ H1 ⊕ Ĥ ⊕Hb︸ ︷︷ ︸
Ḣ1
(4.52)
Ṁ+ =
M22 M23 M24
M32 M33 M34
M42 M43 M44
: H1 ⊕ Ĥ ⊕ H̃b︸ ︷︷ ︸
Ḣ0
→ H1 ⊕ Ĥ ⊕Hb︸ ︷︷ ︸
Ḣ1
(4.53)
Lemma 4.14. Let Π̇ = {Ḣ0 ⊕ Ḣ1, Γ̇0, Γ̇1} be the boundary triplet (4.19)
for T ∗. Moreover, let Ż+(·, λ) ∈ L2
∆[Ḣ0,H] and Ż−(·, λ) ∈ L2
∆[Ḣ1,H] be
operator solutions of (3.3) given by
Ż+(t, λ) = (ξ2(t, λ), ξ3(t, λ), u+(t, λ)) : H1 ⊕ Ĥ ⊕ H̃b → H, λ ∈ C+
(4.54)
Ż−(t, λ) = (ξ2(t, λ), ξ3(t, λ), u−(t, λ)) : H1 ⊕ Ĥ ⊕Hb → H, λ ∈ C−
(4.55)
V. Mogilevskii 253
and let Ṁ+(·) be the operator-function (4.53). Then:
(1) The following equalities hold
Ũ−1Ż+(a, λ) =
(
PH,H0ΓaŻ+(λ)
Γ1
1aŻ+(λ)
)
=
(
S1(λ)
0
)
: Ḣ0 → H0 ⊕H⊥
1 , λ ∈ C+
(4.56)
Ũ−1Ż−(a, λ) =
(
PH,H0ΓaŻ−(λ)
Γ1
1aŻ−(λ)
)
=
(
S∗
2(λ)
0
)
: Ḣ1 → H0 ⊕H⊥
1 , λ ∈ C−.
(4.57)
(2) γ-fields γ̇±(·) of the triplet Π̇ are
γ̇+(λ) = π∆Ż+(λ), λ ∈ C+; γ̇−(λ) = π∆Ż−(λ), λ ∈ C− (4.58)
and the Weyl function of Π̇ coincides with Ṁ+(λ).
(3) If τ is a boundary parameter (4.24), then
(C0(λ)− C1(λ)Ṁ+(λ))
−1 ∈ [Ḣ0]
and
−(τ(λ) + Ṁ+(λ))
−1 = (C0(λ)− C1(λ)Ṁ+(λ))
−1C1(λ), λ ∈ C+.
(4.59)
Proof. (1) It follows from (4.5) and Propositions 4.12, 4.13 that
PH,H0ΓaŻ+(λ) = S1(λ), Γ1
1aŻ+(λ) = 0, λ ∈ C+, (4.60)
and PH,H0ΓaŻ−(λ) = S∗
2(λ), Γ
1
1aŻ−(λ) = 0, λ ∈ C−. This and Lemma
4.6 yield (4.56) and (4.57).
(2) Let γ̇±(λ) be given by (4.58) and let Z±(λ) be the same as in the
proof of Proposition 4.12. Comparing (4.54) and (4.55) with (4.39) and
(4.40) one gets Ż+(λ) =
Z+(λ) ↾ Ḣ0, λ ∈ C+, and Ż−(λ) = Z−(λ) ↾ Ḣ1, λ ∈ C−. Therefore by
(4.38)
Γ′
0Ż+(λ)h0 = h0, h0 ∈ Ḣ0, λ ∈ C+; P1Γ
′
0Ż−(λ)h1 = h1, h1 ∈ Ḣ1, λ ∈ C−
(4.61)
and in view of (4.15) P1Γ
′
0 = (−Γ1
1a, Ṗ1Γ̇
′
0)
⊤ with Ṗ1 := PḢ0,Ḣ1
. This
and (4.14), (4.15) imply that
Γ1
1aŻ+(λ) = 0, λ ∈ C+; Γ1
1aŻ−(λ) = 0, λ ∈ C− (4.62)
Γ̇′
0Ż+(λ) = IḢ0
, λ ∈ C+; Ṗ1Γ̇
′
0Ż−(λ) = IḢ1
, λ ∈ C−. (4.63)
254 Pseudospectral functions
It follows from (4.62) that γ̇+(λ)H0 ⊂ Nλ(T ), γ̇−(λ)H1 ⊂ Nλ(T ) and
(4.63) yields
Γ̇0{γ̇+(λ)h0, λγ̇+(λ)h0} = Γ̇′
0Ż+(λ)h0 = h0, h0 ∈ Ḣ0, λ ∈ C+
Ṗ1Γ̇0{γ̇−(λ)h1, λγ̇−(λ)h1} = Ṗ1Γ̇
′
0Ż−(λ)h1 = h1, h1 ∈ Ḣ1, λ ∈ C−.
Therefore according to definitions (2.12) and (2.13) γ̇±(·) are γ-fields of
Π̇.
Next assume that Ṁ+(·) is given by (4.53). Then in view of (4.42)
and (4.14) Ṁ+(λ) = PH1,Ḣ1
M+(λ) ↾ Ḣ0 and by using (4.48) one obtains
Γ̇′
1Ż+(λ) = PH1,Ḣ1
Γ′
1Z+(λ) ↾ Ḣ0 = PH1,Ḣ1
M+(λ) ↾ Ḣ0 = Ṁ+(λ)
(4.64)
Hence Γ̇1{γ̇+(λ)h0, λγ̇+(λ)h0} = Γ̇′
1Ż+(λ)h0 = Ṁ+(λ)h0, h0 ∈ Ḣ0, λ ∈
C+, and according to definition (2.11) Ṁ+(·) is the Weyl function of Π̇.
Statement (3) follows from [24, Theorem 3.11] and [20, Lemma 2.1].
Theorem 4.15. Let τ be a boundary parameter (4.24), let
C0(λ) = (C0a(λ), Ĉ0(λ), C0b(λ)) : H1 ⊕ Ĥ ⊕ H̃b → Ḣ0 (4.65)
C1(λ) = (C1a(λ), Ĉ1(λ), C1b(λ)) : H1 ⊕ Ĥ ⊕Hb → Ḣ0 (4.66)
be the block representations of C0(λ) and C1(λ) and let
Φ(λ) := (0, C0a(λ), Ĉ0(λ) +
i
2 Ĉ1(λ), −C1a(λ))
: H⊥
1 ⊕H1 ⊕ Ĥ ⊕H1 → Ḣ0. (4.67)
Then for each λ ∈ C+ there exists a unique operator solution vτ (·, λ) ∈
L2
∆[H0,H] of the system (3.3) satisfying the boundary conditions
Γ1
1avτ (λ) = −PH0,H⊥
1
, C0(λ)Γ̇
′
0vτ (λ)− C1(λ)Γ̇
′
1vτ (λ) = Φ(λ)
(4.68)
(here PH0,H⊥
1
is the orthoprojection in H0 onto H⊥
1 in accordance with
decomposition (3.11) of H0).
Proof. Let Z0(·, λ) ∈ L2
∆[H0,H] and Ż+(·, λ) ∈ L2
∆[Ḣ0,H] be operator
solutions of (3.3) given by
Z0(t, λ) = (ξ1(t, λ), ξ2(t, λ), ξ3(t, λ), 0)
: H⊥
1 ⊕H1 ⊕ Ĥ ⊕H1 → H, λ ∈ C+ (4.69)
V. Mogilevskii 255
and (4.54) respectively and let S2(λ) be defined by (4.52). Then in view
of Lemma 4.14,(3) the equality
vτ (t, λ) = Z0(t, λ)+Ż+(t, λ)(C0(λ)−C1(λ)Ṁ+(λ))
−1C1(λ)S2(λ), λ ∈ C+
(4.70)
correctly defines the solution vτ (·, λ) ∈ L2
∆[H0,H] of (3.3). Let us show
that this solution satisfies (4.68).
It follows from (4.5) and Propositions 4.12, 4.13 that
PH,H0ΓaZ0(λ) = m0(λ)− 1
2J0, Γ1
1aZ0(λ) = −PH0,H⊥
1
, λ ∈ C+,
(4.71)
where J0 ∈ [H0] is the operator given by
J0 = PH,H0J ↾ H0 =
0 0 0 0
0 0 0 −IH1
0 0 iI
Ĥ
0
0 IH1 0 0
∈ [H⊥
1 ⊕H1 ⊕ Ĥ ⊕H1︸ ︷︷ ︸
H0
].
(4.72)
Combining (4.70) with the second equalities in (4.71) and (4.60) one gets
the first equality in (4.68). Next, by (4.63) and (4.64)
(C0(λ)Γ̇
′
0 − C1(λ)Γ̇
′
1)vτ (λ) = (C0(λ)Γ̇
′
0 − C1(λ)Γ̇
′
1)Z0(λ)
+ (C0(λ)Γ̇
′
0 − C1(λ)Γ̇
′
1)Ż+(λ)(C0(λ)− C1(λ)Ṁ+(λ))
−1C1(λ)S2(λ)
= C0(λ)Γ̇
′
0Z0(λ) + C1(λ)(S2(λ)− Γ̇′
1Z0(λ)).
Moreover, by (4.28)–(4.33) and (4.43)–(4.45) one has
Γ̇′
0Z0(λ) =
0 IH1 0 0
0 0 I
Ĥ
0
0 0 0 0
,
Γ̇′
1Z0(λ) =
M21(λ) M21(λ) M23(λ) 0
M31(λ) M32(λ) M33(λ) 0
M41(λ) M42(λ) M43(λ) 0
and hence S2(λ) − Γ̇′
1Z0(λ) =
0 0 0 −IH1
0 0 i
2IĤ 0
0 0 0 0
. This and (4.65),
256 Pseudospectral functions
(4.66) imply that
C0(λ)Γ̇0vτ (λ)− C1(λ)Γ̇1vτ (λ)
= (C0a(λ), Ĉ0(λ), C0b(λ))
0 IH1 0 0
0 0 I
Ĥ
0
0 0 0 0
+ (C1a(λ), Ĉ1(λ), C1b(λ))
0 0 0 −IH1
0 0 i
2IĤ 0
0 0 0 0
= Φ(λ).
Thus the second equality in (4.68) is valid. Finally uniqueness of vτ (·, λ)
is implied by uniqueness of the solution of the problem (4.25), (4.26) (see
Theorem 4.11).
4.3. m-functions
Let τ be a boundary parameter (4.24), let vτ (·, λ) ∈ L2
∆[H0,H] be
the operator solution of (3.3) defined in Theorem 4.15 and let J0 be the
operator (4.72).
Definition 4.16. The operator function mτ (·) : C+ → [H0] defined by
mτ (λ) = PH,H0Γavτ (λ) +
1
2J0, λ ∈ C+ (4.73)
is called the m-function corresponding to the boundary parameter τ or,
equivalently, to the boundary value problem (4.25), (4.26).
In the following theorem we provide a description of all m-functions
immediately in terms of the boundary parameter τ .
Theorem 4.17. Let the assumptions (A1)–(A3) after Proposition 4.4 be
satisfied and let Ḣ0 and Ḣ1 be finite-dimensional Hilbert spaces (4.11).
Assume also that M+(·) is the operator-function defined by (4.42)–(4.45)
and m0(·), S1(·), S2(·) and Ṁ+(·) are the operator-functions (4.50)–
(4.53). Then:
(1) m0(·) is the m-function corresponding to the boundary parameter
τ0 = {IḢ0
, 0Ḣ1,Ḣ0
};
(2) for every boundary parameter τ of the form (4.24) the correspond-
ing m-function mτ (·) admits the representation
mτ (λ) = m0(λ) + S1(λ)(C0(λ)− C1(λ)Ṁ+(λ))
−1C1(λ)S2(λ), λ ∈ C+.
(4.74)
Proof. Applying the operator PH,H0Γa to the equality (4.70) and tak-
ing the first equalities in (4.71) and (4.60) into account one gets (4.74).
Statement (1) of the theorem is immediate from (4.74).
V. Mogilevskii 257
Proposition 4.18. The m-function mτ (·) belongs to the class R[H0] and
satisfies
Immτ (λ) ≥
∫
I
v∗τ (t, λ)∆(t)vτ (t, λ) dt, λ ∈ C+. (4.75)
If N+ = N− and τ is a self-adjoint boundary parameter, then the in-
equality (4.75) turns into the equality.
Proof. It follows from (4.74) that mτ (·) is holomorphic in C+. Moreover,
one can prove inequality (4.75) in the same way as similar inequalities
(5.10) in [1] and (4.66) in [25]. Therefore mτ (·) ∈ R[H0].
4.4. Generalized resolvents and characteristic matrices
In the sequel we denote by Y
Ũ
(·, λ) the [H]-valued operator solution
of (3.3) satisfying Y
Ũ
(a, λ) = Ũ , λ ∈ C.
The following theorem is well known (see e.g. [5, 9, 33]).
Theorem 4.19. For each generalized resolvent R(λ) of Tmin there exists
an operator-function Ω(·) ∈ R[H] (the characteristic matrix of R(λ))
such that for any f̃ ∈ H and λ ∈ C+
R(λ)f̃ = π∆
(∫
I
Y
Ũ
(x, λ)(Ω(λ) + 1
2 sgn(t− x)J)Y ∗
Ũ
(t, λ)∆(t)f(t) dt
)
,
f ∈ f̃ . (4.76)
Proposition 4.20. Let τ be a boundary parameter (4.24) and let Rτ (λ)
be the corresponding generalized resolvent of T (and hence of Tmin) in
accordance with Theorem 4.11. Moreover, let PH0,H⊥
1
and IH⊥
1 ,H0
be the
orthoprojection in H0 onto H⊥
1 and the embedding operator of H⊥
1 into
H0 respectively (see decomposition (3.11) of H0). Then the equality
Ω(λ) =
(
mτ (λ) −1
2IH⊥
1 ,H0
−1
2PH0,H⊥
1
0
)
: H0 ⊕H⊥
1︸ ︷︷ ︸
H
→ H0 ⊕H⊥
1︸ ︷︷ ︸
H
, λ ∈ C+
(4.77)
defines a characteristic matrix Ω(·) of Rτ (λ).
Proof. Assume that γ̇±(·) are γ-fields and Ṁ+(·) is the Weyl function of
the boundary triplet Π̇ = {Ḣ0 ⊕ Ḣ1, Γ̇0, Γ̇1} for T ∗ defined in Proposi-
tion 4.9. Moreover let
Bτ (λ) := −(τ(λ) + Ṁ+(λ))
−1 = (C0(λ)−C1(λ)Ṁ+(λ))
−1C1(λ), λ ∈ C+
(4.78)
258 Pseudospectral functions
(see (4.59)). Then according to [24, Theorem 3.11] the Krein type for-
mula for generalized resolvents
R(λ) = Rτ (λ) = (A0 − λ)−1 + γ̇+(λ)Bτ (λ)γ̇
∗
−(λ), λ ∈ C+ (4.79)
holds with the maximal symmetric extension A0 of T given by
A0 := ker Γ̇0
= {π̃∆{y, f} : {y, f} ∈ Tmax, Γ
1
1ay = 0, Γ2
1ay = 0, Γ̂ay = Γ̂by, Γ0by = 0}.
According to [25, (4.36)] for each f̃ ∈ H and λ ∈ C+
(A0 − λ)−1f̃
= π∆
(∫
I
Y
Ũ
(x, λ)(Ω0(λ) +
1
2 sgn(t− x)J)Y ∗
Ũ
(t, λ)∆(t)f(t)dt
)
, (4.80)
where f(·) ∈ f̃ and Ω0(λ) is the operator function defined in [25, (4.30)]
(actually (4.80) is proved in [25] for definite systems but the proof is
suitable for the case of a θ-definite system as well). One can easily verify
that Ω0(λ) admits the representation
Ω0(λ) =
(
m0(λ) −1
2IH⊥
1 ,H0
−1
2PH0,H⊥
1
0
)
: H0 ⊕H⊥
1 → H0 ⊕H⊥
1 , λ ∈ C+
(4.81)
with m0(λ) given by (4.50). Next, Ż±(t, λ) = Y
Ũ
(t, λ)Ũ−1Ż±(a, λ) and
in view of the second equality in (4.58) and [1, Lemma 3.3] one has
γ̇∗−(λ)f̃ = ∫
I
Ż∗
−(t, λ)∆(t)f(t) dt
= ∫
I
(Ũ−1Ż−(a, λ))∗Y ∗
Ũ
(t, λ)∆(t)f(t) dt, f(·) ∈ f̃ .
This and the first equality in (4.58) imply that for any f̃ ∈ H and λ ∈ C+
γ̇+(λ)Bτ (λ)γ̇
∗
−(λ)f̃
= π∆ ∫
I
Y
Ũ
(·, λ)(Ũ−1Ż+(a, λ))Bτ (λ)(Ũ
−1Ż−(a, λ))∗Y ∗
Ũ
(t, λ)∆(t)f(t) dt
= π∆ ∫
I
Y
Ũ
(·, λ)Ω(λ)Y ∗
Ũ
(t, λ)∆(t)f(t) dt, f(·) ∈ f̃ ,
where
Ω(λ) = (Ũ−1Ż+(a, λ))Bτ (λ)(Ũ
−1Ż−(a, λ))∗ =
(
S1(λ)
0
)
Bτ (λ) (S2(λ), 0)
=
(
S1(λ)Bτ (λ)S2(λ) 0
0 0
)
V. Mogilevskii 259
(here we made use of (4.56) and (4.57)). Combining these relations with
(4.79) and (4.80) one obtains the equality (4.76) with
Ω(λ) = Ω0(λ) + Ω(λ) =
(
m0(λ) + S1(λ)Bτ (λ)S2(λ) −1
2IH⊥
1 ,H0
−1
2PH0,H⊥
1
0
)
.
Hence Ω(λ) is a characteristic matrix of Rτ (λ) and in view of (4.74) and
(4.78) the equality (4.77) is valid.
5. Parametrization of pseudospectral and spectral func-
tions
As before we suppose in this section (unless otherwise stated)the as-
sumptions (A1)–(A3) specified after Proposition 4.4.
Let T be a symmetric relation (3.12). Then according to Theorem
4.11 the boundary value problem (4.25), (4.26) induces parametrizations
R(λ) = Rτ (λ), T̃ = T̃τ and F (·) = Fτ (·) of all generalized resolvents
R(λ), exit space extensions T̃ ∈ S̃elf(T ) and spectral functions F (·) of T
respectively by means of the boundary parameter τ . Here T̃τ (∈ S̃elf(T ))
is the extension of T generating Rτ (λ) and Fτ (·) is the respective spectral
function of T .
Definition 5.1. Let Ṁ+ = Ṁ+(λ) be given by (4.53). A boundary
parameter τ of the form (4.24) is called admissible if
lim
y→+∞
1
iyPḢ0,Ḣ1
(C0(iy)− C1(iy)Ṁ+(iy))
−1C1(iy) = 0, (5.1)
lim
y→+∞
1
iyṀ+(iy)(C0(iy)− C1(iy)Ṁ+(iy))
−1C0(iy) ↾ Ḣ1 = 0. (5.2)
Proposition 5.2. An extension T̃ = T̃ τ belongs to Self0(T ) if and only
if the boundary parameter τ is admissible. Therefore the set of admissible
boundary parameters is not empty.
Proof. According to Lemma 4.14, (2) Ṁ+(·) is the Weyl function of the
boundary triplet Π̇ for T ∗. Therefore the required result follows from [26,
Theorem 2.15].
In the following with the operator Ũ from assumption (A2) we asso-
ciate the operator U = Uθ ∈ [H0,H] given by U = Ũ ↾ H0. Moreover,
we denote by ϕU (·, λ) the [H0,H]-valued operator solution of (3.3) with
ϕU (a, λ) = U . Clearly kerU = {0} and UH0 = θ.
260 Pseudospectral functions
Theorem 5.3. Let τ be an admissible boundary parameter, let F (·) =
Fτ (·) be the corresponding spectral function of T and let mτ (·) be the
m-function (4.73). Then there exists a unique pseudospectral function
σ(·) = στ (·) of the system (3.3) (with respect to U ∈ [H0,H]) satisfying
(3.19). This pseudospectral function is defined by the Stieltjes inversion
formula
στ (s) = lim
δ→+0
lim
ε→+0
1
π
∫ s−δ
−δ
Immτ (u+ iε) du. (5.3)
Proof. Assume that Ω(·) ∈ R[H] is the characteristic matrix (4.77) of
Rτ (λ) and Σ(·) : R → [H] is the distribution function defined by
Σ(s) = lim
δ→+0
lim
ε→+0
1
π
∫ s−δ
−δ
ImΩ(u+ iε) du.
Using (4.76) and the Stieltjes–Livs̆ic formula one proves as in [9, 33] the
equality
((F (β)− F (α))f̃ , f̃) =
∫
[α,β)
(dΣ(s)f̂0(s), f̂0(s)), f̃ ∈ Hb,
−∞ < α < β <∞ (5.4)
with the function f̂0 : R → H defined for each f̃ ∈ Hb by f̂0(s) =∫
I
Y ∗
ũ (t, s)∆(t)f(t) dt, f(·) ∈ f̃ . Let f̃ ∈ Hb, let
f̂(s) =
∫
I
ϕ∗
u(t, s)∆(t)f(t) dt, f(·) ∈ f̃ (5.5)
and let σ(·) = στ (·) be the distribution function (5.3). Since ϕu(t, λ) =
Yũ(t, λ) ↾ H0, it follows that f̂(s) = PH,H0 f̂0(s). Moreover, by (4.77) one
has
Σ(s) =
(
σ(s) 0
0 0
)
: H0 ⊕H⊥
1 → H0 ⊕H⊥
1 .
This and (5.4) yield the equality (3.19). Next by using (3.19) and Propo-
sition 5.2 one proves that σ(·) is a pseudospectral function (with respect
to U) in the same way as in [26, Theorem 3.20] and [27, Theorem 5.4].
Let us prove that σ(·) = στ (·) is a unique pseudospectral function sat-
isfying (3.19) (we give only the sketch of the proof because it is similar to
that of the alike result in [27, Theorem 5.4]). Let σ̃(·) be a pseudospec-
tral function (with respect to U) such that (3.19) holds with σ̃(·) instead
V. Mogilevskii 261
of σ(·). Then according to [10] there exists a scalar measure µ on Borel
sets in R and functions Ψj(·) : R → [H0], j ∈ {1, 2}, such that
σ(β)− σ(α) =
∫
δ
Ψ1(s) dµ(s), σ̃(β)− σ̃(α) =
∫
δ
Ψ2(s) dµ(s), δ = [α, β).
(5.6)
Let Ψ(s) := Ψ1(s)−Ψ2(s) and let µ0 be the Lebesgue measure on Borel
sets in I. Denote also by G the set of all functions f̂(·) : R → H0
admitting the representation (5.5) with some f̃ ∈ Hb. As in [27, Theorem
5.4] one proves that for each f̂ ∈ G there is a Borel set C
f̂
⊂ R such that
µ(R \ C
f̂
) = 0 and µ0({t ∈ I : ∆(t)ϕU (t, s)Ψ(s)f̂(s) 6= 0}) = 0, s ∈ C
f̂
.
(5.7)
Let s ∈ C
f̂
and let y = y(t) = ϕU (t, s)Ψ(s)f̂(s). Then y is a solution
of the system (3.3) with λ = s and by (5.7) ∆(t)y(t) = 0 (µ0 a.e. on
I). Hence y ∈ N . Moreover, y(a) = UΨ(s)f̂(s) ∈ θ. Since system is
θ-definite, this implies that y = 0 and, consequently, Ψ(s)f̂(s) = 0. Thus
for any f̂ ∈ G there exists a Borel set C
f̂
⊂ R such that
µ(R \ C
f̂
) = 0 and Ψ(s)f̂(s) = 0, s ∈ C
f̂
. (5.8)
Next we prove the following statement:
(S) for any s ∈ R and h ∈ H0 there is f̂(·) ∈ G such that f̂(s) = h.
Indeed, let s ∈ R, h′ ∈ H0 and (f̂(s), h′) = 0 for any f̂(·) ∈ G.
Put y = y(t) = ϕU (t, s)h
′. Then for any β ∈ I one has f̂β(·) :=∫
[a,β]
ϕ∗
U (t, ·)∆(t)y(t) dt ∈ G and, consequently,
0 = (f̂β(s), h
′) =
∫
[a,β]
(ϕ∗
U (t, s)∆(t)y(t), h′) dt
=
∫
[a,β]
(∆(t)y(t), y(t)) dt, β ∈ I.
Hence y ∈ N . Moreover, y(a) = Uh′ ∈ θ and θ-definiteness of the system
implies that y = 0. Therefore h′ = 0, which proves statement (S).
Next by using (5.8) and statement (S) one proves the equality Ψ(s) =
0 (µ-a.e. on R) in the same way as in [27, Theorem 5.4]. Thus Ψ1(s) =
Ψ2(s) (µ-a.e. on R) and by (5.6) σ̃(s) = σ(s).
Corollary 5.4. (1) Let the assumption (A1) from Section 4.1 be sat-
isfied. Then the set of pseudospectral functions (with respect to Kθ ∈
[H′
0,H]) is not empty.
262 Pseudospectral functions
(2) Let system (3.3) be definite, let N− ≤ N+ and let θ be a subspace
in H. Then the set of pseudospectral functions (with respect to Kθ ∈
[H′
0,H]) is not empty if and only if θ× ∈ Sym(H).
Proof. Statement (1) is immediate from Proposition 5.2 and Theorem
5.3. Statement (2) follows from statement (1), Remark 4.2 and Proposi-
tion 3.16.
A parametrization of all pseudospectral functions σ(·) (with respect
to U ∈ [H0,H]) immediately in terms of a boundary parameter τ is given
by the following theorem.
Theorem 5.5. Let the assumptions be the same as in Theorem 4.17.
Then the equality
mτ (λ) = m0(λ) + S1(λ)(C0(λ)− C1(λ)Ṁ+(λ))
−1C1(λ)S2(λ), λ ∈ C+
(5.9)
together with formula (5.3) establishes a bijective correspondence σ(s) =
στ (s) between all admissible boundary parameters τ defined by (4.24)
and all pseudospectral functions σ(·) of the system (3.3) (with respect to
U ∈ [H0,H]).
The proof of Theorem 5.5 is based on Theorems 5.3, 4.17 and Propo-
sitions 3.14, 5.2. We omit this proof because it is similar to that of
Theorem 5.7 in [27].
The following theorem directly follows from Theorem 5.3 and Propo-
sitions 3.14, 4.4.
Theorem 5.6. Let the assumptions (A1) and (A2) from Section 4.1 be
satisfied. Then there is a one to one correspondence σ(·) = σ
T̃
(·) between
all extensions T̃ ∈ S̃elf0(T ) and all pseudospectral functions σ(·) of the
system (3.3) (with respect to U ∈ [H0,H]). This correspondence is given
by the equality (3.19), where F (·) is a spectral function of T generated
by T̃ . Moreover, the operators T̃0 (the operator part of T̃ ) and Λσ are
unitarily equivalent and hence they have the same spectral properties. In
particular this implies that the spectral multiplicity of T̃0 does not exceed
dimH0.
Corollary 5.7. Let under the assumptions (A1)–(A3) τ be an admissible
boundary parameter, let σ(·) = στ (·) be a pseudospectral function (with
respect to U) and let V0,σ(= Vσ ↾ H0) be the corresponding isometry
from H0 to L2(σ;H0) . Then V0,σ is a unitary operator if and only if
the parameter τ is self-adjoint. If this condition is satisfied, then the
boundary conditions (4.27) defines an extension T̃ τ ∈ Self0(T ) and the
V. Mogilevskii 263
operators T̃0,τ (the operator part of T̃ τ ) and Λσ are unitarily equivalent
by means of V0,σ
Proof. The first statement is a consequence of Proposition 3.14 and Theo-
rem 4.11. The second statement is implied by Theorems 4.11 and 5.6.
The criterion which enables one to describe all pseudospectral func-
tions in terms of an arbitrary (not necessarily admissible) boundary pa-
rameter is given in the following theorem.
Theorem 5.8. The following statements are equivalent:
(1) each boundary parameter τ is admissible;
(2) lim
y→+∞
Ṁ+(iy) ↾ Ḣ1 = 0 and
lim
y→+∞
y
(
Im(Ṁ+(iy)h, h)Ḣ0
+ 1
2 ||Ṗ2h||2
)
= +∞,
where h ∈ Ḣ0, h 6= 0 and Ṗ2 is the orthoprojection in Ḣ0 onto Ḣ2 :=
Ḣ0 ⊖ Ḣ1;
(3) mulT = mulT ∗, i.e., the condition (C2) in Assertion 4.3 is ful-
filled;
(4) statement of Theorem 5.5 holds for arbitrary boundary parameters
τ .
Proof. Proposition 5.2 and (2.5) yield the equivalence (1)⇔ (3). Since by
Lemma 4.14, (2) Ṁ+(·) is the Weyl function of the boundary triplet Π̇,
the equivalence (2)⇔ (3) is implied by [24, Theorem 4.6]. The equivalence
(1) ⇔(4) follows from Theorem 5.5.
Combining the results of this section with Proposition 3.11 we get the
following theorem.
Theorem 5.9. Let the assumptions (A1) and (A2) be satisfied. Then the
set of spectral functions of the system (3.3) (with respect to U ∈ [H0,H])
is not empty if and only if mulT = {0} or equivalently if and only if the
condition (C1) in Assertion 4.3 is fulfilled. If this condition is satisfied,
then the sets of spectral and pseudospectral functions of the system (3.3)
coincide and hence Theorems 5.5, 5.6, 5.8 and Corollary 5.7 are valid for
spectral functions (instead of pseudospectral ones). In this case T̃0, T̃0,τ
and V0,σ in Theorem 5.6 and Corollary 5.7 should be replaced with T̃ , T̃τ
and Vσ respectively. Moreover, in this case statement (3) in Theorem 5.8
takes the following form:
(3’) mulT ∗ = {0}, i.e., the condition (C3) in Assertion 4.3 is fulfilled.
264 Pseudospectral functions
Remark 5.10. Assume that N− ≤ N+ and θ is a subspace in H such
that θ× ∈ Sym(H) and system (3.3) is θ-definite. Moreover, let H′
0 be
a subspace in H and let Kθ ∈ [H′
0,H] be an operator with kerKθ =
{0} and KθH
′
0 = θ. It follows from Proposition 3.12 and Remark 3.13
that Theorems 5.5, 5.6,5.8, 5.9 and Corollary 5.7 are valid, with some
corrections, for pseudospectral and spectral functions σ(·) with respect
to Kθ in place of U . We leave to the reader the precise formulation of
the specified results.
6. The case of the minimally possible dim θ. Spectral
functions of the minimal dimension
It follows from Lemma 3.1, (1) that the minimally possible dimension
of the subspace θ ⊂ H satisfying the assumption (A1) in Section 4.1 is
dim θ = ν + ν̂. (6.1)
If θ satisfies (A1) and (6.1) then the previous results become essentially
simpler. Namely, in this case the subspace H0 from assumption (A2)
satisfies dimH0 = dim(H ⊕ Ĥ) and hence H1 = {0}, H⊥
1 = H and
H0 = H ⊕ Ĥ. (6.2)
Therefore the assumption (A2) in Section 4.1 takes the following form:
(A2′) H0 is the subspace (6.2), Ũ and Γa are the same as in the
assumption (A2) and
Γa = (Γ0a, Γ̂a,Γ1a)
⊤ : dom Tmax → H ⊕ Ĥ ⊕H (6.3)
is the block representation of Γa.
Below we suppose (unless otherwise is stated) the following assump-
tion (Amin), which is equivalent to the assumptions (A1) - (A3) and the
equality (6.1):
(Amin) In addition to (A1) the equality (6.1) holds and the assump-
tions (A2′) and (A3) are satisfied.
Under this assumption the equalities (4.11) take the form
Ḣ0 = Ĥ ⊕ H̃b, Ḣ1 = Ĥ ⊕Hb (6.4)
and a boundary parameter is the same as in definition 4.10.
Theorem 6.1. Let τ be a boundary parameter (4.24) and let
C0(λ) = (Ĉ0(λ), C0b(λ)) : Ĥ ⊕ H̃b → Ḣ0,
C1(λ) = (Ĉ1(λ), C1b(λ)) : Ĥ ⊕Hb → Ḣ0
V. Mogilevskii 265
be the block representations of C0(λ) and C1(λ). Then for each λ ∈ C+
there exists a unique pair of operator solutions ξτ (·, λ) ∈ L2
∆[H,H] and
ξ̂τ (·, λ) ∈ L2
∆[Ĥ,H] of the system (3.3) satisfying the boundary conditions
Γ1aξτ (λ) = −IH , (6.5)
[(iĈ0(λ)− 1
2 Ĉ1(λ))Γ̂a + C0b(λ)Γ0b − (iĈ0(λ) +
1
2 Ĉ1(λ))Γ̂b
+C1b(λ)Γ1b]ξτ (λ) = 0, (6.6)
Γ1aξ̂τ (λ) = 0, (6.7)
[(iĈ0(λ)− 1
2 Ĉ1(λ))Γ̂a + C0b(λ)Γ0b
−(iĈ0(λ) +
1
2 Ĉ1(λ))Γ̂b + C1b(λ)Γ1b]ξ̂τ (λ) = Ĉ0(λ) +
i
2 Ĉ1(λ). (6.8)
Proof. Let vτ (·, λ) ∈ L2
∆[H0,H] be the solution of (3.3) defined in theo-
rem 4.15 and let
vτ (t, λ) = (ξτ (t, λ), ξ̂τ (t, λ)) : H ⊕ Ĥ → H (6.9)
be the block representation of vτ (t, λ). Then the first condition in (4.68)
takes the form Γ1a(ξτ (λ), ξ̂τ (λ)) = (−IH , 0), which is equivalent to (6.5)
and (6.7). Moreover, (4.12), (4.13) and (4.67) take the form
Γ̇′
0 = (i(Γ̂a − Γ̂b), Γ0b)
⊤, Γ̇′
1 = (12(Γ̂a + Γ̂b), −Γ1b)
⊤,
Φ(λ) = (0, Ĉ0(λ) +
i
2 Ĉ1(λ)).
Therefore the second condition in (4.68) is equivalent to (6.6) and (6.8).
Now the required statement is implied by Theorem 4.15.
It follows from (6.2), (6.3) and (4.72) that PH,H0Γa = (Γ0a, Γ̂a)
⊤ and
J0 =
(
0 0
0 iI
Ĥ
)
. This and (4.73) imply that in the case (6.1) (i.e., under
the assumption (Amin)) the m-function mτ (·) can be defined as
mτ (λ) =
(
Γ0aξτ (λ) Γ0aξ̂τ (λ)
Γ̂aξτ (λ) Γ̂aξ̂τ (λ) +
i
2IĤ
)
: H ⊕ Ĥ︸ ︷︷ ︸
H0
→ H ⊕ Ĥ︸ ︷︷ ︸
H0
, λ ∈ C+.
The following proposition is implied by Proposition 4.12.
Proposition 6.2. For any λ ∈ C+ there exists a unique collection of
operator solutions ξ0(·, λ) ∈ L2
∆[H,H], ξ̂0(·, λ) ∈ L2
∆[Ĥ,H] and u+(·, λ) ∈
L2
∆[H̃b,H] of the system (3.3) satisfying the boundary conditions
Γ1aξ0(λ) = −IH , Γ̂aξ0(λ) = Γ̂bξ0(λ), Γ0bξ0(λ) = 0,
Γ1aξ̂0(λ) = 0, i(Γ̂a − Γ̂b)ξ̂0(λ) = I
Ĥ
, Γ0bξ̂0(λ) = 0,
Γ1au+(λ) = 0, Γ̂au+(λ) = Γ̂bu+(λ), Γ0bu+(λ) = IH̃b
.
266 Pseudospectral functions
If the assumption (Amin) is satisfied, then the operator functionM+(·)
from Proposition 4.13 takes the form
M+(λ) =
M11(λ) M12(λ) M13(λ)
M21(λ) M22(λ) M23(λ)
M31(λ) M32(λ) M33(λ)
: H ⊕ Ĥ ⊕ H̃b → H ⊕ Ĥ ⊕Hb,
where λ ∈ C+ and
M11(λ) = Γ0aξ0(λ), M12(λ) = Γ0aξ̂0(λ), M13(λ) = Γ0au+(λ),
M21(λ) = Γ̂aξ0(λ), M22(λ) = Γ̂aξ̂0(λ) +
i
2IĤ , M23(λ) = Γ̂au+(λ),
M31(λ) = −Γ1bξ0(λ), M32(λ) = −Γ1bξ̂0(λ), M33(λ) = −Γ1bu+(λ).
Moreover, the operator functions m0(·), S1(·), S2(·) and Ṁ+(·) in The-
orem 5.5 take the following simpler form (cf. (4.50)-(4.53)):
m0(λ) =
(
M11(λ) M12(λ)
M21(λ) M22(λ)
)
: H ⊕ Ĥ︸ ︷︷ ︸
H0
→ H ⊕ Ĥ︸ ︷︷ ︸
H0
, λ ∈ C+
S1(λ) =
(
M12(λ) M13(λ)
M22(λ)− i
2IĤ M23(λ)
)
: Ĥ ⊕ H̃b︸ ︷︷ ︸
Ḣ0
→ H ⊕ Ĥ︸ ︷︷ ︸
H0
, λ ∈ C+
S2(λ) =
(
M21(λ) M22(λ) +
i
2IĤ
M31(λ) M32(λ)
)
: H ⊕ Ĥ︸ ︷︷ ︸
H0
→ Ĥ ⊕Hb︸ ︷︷ ︸
Ḣ1
, λ ∈ C+
Ṁ+(λ) =
(
M22(λ) M23(λ)
M32(λ) M33(λ)
)
: Ĥ ⊕ H̃b︸ ︷︷ ︸
Ḣ0
→ Ĥ ⊕Hb︸ ︷︷ ︸
Ḣ1
, λ ∈ C+.
In the following theorem we characterize spectral functions of the minimal
dimension.
Theorem 6.3. Let system (3.3) be definite (see Definition 3.15) and let
N− ≤ N+. Then the following statements are equivalent:
(i) mulTmin = {0}, i.e., the condition (C0) in Assertion 4.3 is ful-
filled;
(ii) The set of spectral functions of the system is not empty, i.e., there
exist subspaces θ and H′
0 in H and a spectral function σ(·) of the system
(with respect to Kθ ∈∈ [H′
0,H]).
If the statement (i) holds, then the dimension nσ of each spectral
function σ(·) (see Definition 3.10) satisfies
ν + ν̂ ≤ nσ ≤ n (6.10)
and there exists a spectral function σ(·) with the minimally possible di-
mension nσ = ν + ν̂.
V. Mogilevskii 267
Proof. Assume statement (i). Then by Lemma 3.4 there exists a subspace
θ ⊂ H such that θ× ∈ Sym(H), dim θ = ν + ν̂ and the relation T of the
form (3.12) satisfies mulT = {0}. Therefore by Corollary 5.4, (2) and
Proposition 3.11 there exists a spectral function σ(·) (with respect to
Kθ). Moreover, nσ(= dim θ) = ν + ν̂.
Next assume that θ is a subspace in H and σ(·) is a spectral func-
tion (with respect to Kθ). Since the system is definite, it follows from
Proposition 3.16 that θ× ∈ Sym(H). Therefore by Lemma 3.1, (1)
nσ(= dim θ) ≥ ν + ν̂, which yields (6.10).
Conversely, let statement (ii) holds. If σ(·) is a spectral function
(with respect to Kθ), then according to Proposition 3.11 mulT = {0}.
This and the obvious inclusion mulTmin ⊂ mulT yield statement (i).
References
[1] S. Albeverio, M.M. Malamud, and V.I.Mogilevskii, On Titchmarsh-Weyl func-
tions and eigenfunction expansions of first-order symmetric systems, Integr. Equ.
Oper. Theory, 77 (2013), 303–354.
[2] D. Z. Arov, H. Dym, Bitangential direct and inverse problems for systems of inte-
gral and differential equations, Encyclopedia of mathematics and its applications,
Cambridge University Press, Cambridge, 2012.
[3] F. V. Atkinson, Discrete and continuous boundary problems, Academic Press,
New York, 1963.
[4] V. M. Bruk, Extensions of symmetric relations, Math. Notes, 22 (1977), No. 6,
953–958.
[5] V. M. Bruk, Linear relations in a space of vector functions, Math. Notes, 24
(1978), No. 4, 767–773.
[6] V. A. Derkach, S. Hassi, M.M. Malamud, and H.S.V. de Snoo, Boundary relations
and their Weyl families, Trans. Amer. Math. Soc., 358 (2006), 5351–5400.
[7] V. A. Derkach, M. M. Malamud, Generalized resolvents and the boundary value
problems for Hermitian operators with gaps, J. Funct. Anal., 95 (1991), 1–95.
[8] A. Dijksma, H. Langer, and H. S. V. de Snoo, Hamiltonian systems with eigen-
value depending boundary conditions, Oper. Theory Adv. Appl. 35 (1988), 37–83.
[9] A. Dijksma, H. Langer, and H. S. V. de Snoo, Eigenvalues and pole functions
of Hamiltonian systems with eigenvalue depending boundary conditions, Math.
Nachr., 161 (1993), 107–153.
[10] N. Dunford and J. T. Schwartz, Linear operators. Part2. Spectral theory, Inter-
science Publishers, New York–London, 1963.
268 Pseudospectral functions
[11] I. Gohberg and M. G. Krein, Theory and applications of Volterra operators in
Hilbert space, Transl. Math. Monographs, 24, Amer. Math. Soc., Providence,
R.I., 1970.
[12] V. I. Gorbachuk and M. L. Gorbachuk, Boundary problems for differential-
operator equations, Kluver Acad. Publ., Dordrecht–Boston–London, 1991.
[13] D. B. Hinton and A. Schneider, On the Titchmarsh-Weyl coefficients for singular
S-Hermitian systems I, Math. Nachr., 163 (1993), 323–342.
[14] I. S. Kats, Linear relations generated by the canonical differential equation of
phase dimension 2, and eigenfunction expansion, St. Petersburg Math. J., 14
(2003), 429–452.
[15] V. I. Kogan and F. S. Rofe-Beketov, On square-integrable solutions of symmetric
systems of differential equations of arbitrary order, Proc. Roy. Soc. Edinburgh
Sect. A, 74 (1974/75), 5–40.
[16] H. Langer and B. Textorius, Spectral functions of a symmetric linear relation with
a directing mapping, I, Proc. Roy. Soc. Edinburgh Sect. A, 97 (1984), 165–176.
[17] H. Langer and B. Textorius, Spectral functions of a symmetric linear relation with
a directing mapping, II, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 111–124.
[18] M. Lesch and M.M. Malamud, On the deficiency indices and self-adjointness of
symmetric Hamiltonian systems, J. Differential Equations, 189 (2003), 556–615.
[19] M. M. Malamud, On the formula of generalized resolvents of a nondensely defined
Hermitian operator, Ukr. Math. Zh., 44 (1992), No. 12, 1658–1688.
[20] M. M. Malamud and V. I. Mogilevskii, Krein type formula for canonical resolvents
of dual pairs of linear relations, Methods Funct. Anal. Topology, 8 (2002), No. 4,
72–100.
[21] V. I. Mogilevskii, Boundary triplets and Krein type resolvent formula for sym-
metric operators with unequal defect numbers, Methods Funct. Anal. Topology,
12 (2006), No. 3, 258–280.
[22] V. I. Mogilevskii, Boundary pairs and boundary conditions for general (not nec-
essarily definite) first-order symmetric systems with arbitrary deficiency indices,
Math. Nachr., 285 (2012), No. 14–15, 1895–1931.
[23] V. Mogilevskii, Minimal spectral functions of an ordinary differential operator,
Proc. Edinb. Math. Soc., 55 (2012), No. 3, 731–769.
[24] V. I. Mogilevskii, On exit space extensions of symmetric operators with appli-
cations to first order symmetric systems, Methods Funct. Anal. Topology, 19
(2013), No. 3, 268–292.
[25] V. I. Mogilevskii, On generalized resolvents and characteristic matrices of first-
order symmetric systems, Methods Funct. Anal. Topology, 20 (2014), No. 4,
328–348.
V. Mogilevskii 269
[26] V. I. Mogilevskii, On spectral and pseudospectral functions of first- order sym-
metric systems, Ufa Math. Journal, 7 (2015), No. 2, 115–136.
[27] V. I. Mogilevskii, Spectral and pseudospectral functions of Hamiltonian systems:
development of the results by Arov-Dym and Sakhnovich, Methods Funct. Anal.
Topology, 21 (2015), No. 4, 370–402.
[28] V. Mogilevskii, On eigenfunction expansions of first-order symmetric systems and
ordinary differential operators of an odd order, Integr. Equ. Oper. Theory, 82
(2015), 301–337.
[29] M. A. Naimark, Linear differential operators, vol. 1 and 2, Harrap, London, 1967.
[30] B. C. Orcutt, Canonical differential equations, Dissertation, University of Vir-
ginia, 1969.
[31] A.L. Sakhnovich, Spectral functions of a canonical system of order 2n, Mat. Sb.
181 (1990), No. 11, 1510–1524 [in Russian]; transl.: Math. USSR-Sb., 71 (1992),
No. 2, 355–369.
[32] A.L. Sakhnovich, L.A. Sakhnovich, and I.Ya. Roitberg, Inverse problems and
nonlinear evolution equations. Solutions, Darboux matrices and Weyl-Titchmarsh
functions, De Gruyter Studies in Mathematics 47. De Gruyter, Berlin, 2013.
[33] A.V. S̆traus, On generalized resolvents and spectral functions of differential oper-
ators of an even order, Izv. Akad. Nauk. SSSR, Ser.Mat., 21, (1957), 785–808.
Contact information
Vadim Mogilevskii Poltava V.G. Korolenko
National Pedagogical University,
Poltava, Ukraine
E-Mail: vadim.mogilevskii@gmail.com
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