On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the InЇnite-Dimensional Case
We define the families of maximal and minimal relations generated by an integral equation with a Nevanlinna operator measure in the infinite-dimensional case and prove their holomorphic property. We show that if the restrictions of maximal relations are continuously invertible, then the operators in...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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Bruk, V.M. 2016-10-05T19:23:16Z 2016-10-05T19:23:16Z 2014 On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the InЇnite-Dimensional Case / V.M. Bruk // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 2. — С. 163-188. — Бібліогр.: 24 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106790 We define the families of maximal and minimal relations generated by an integral equation with a Nevanlinna operator measure in the infinite-dimensional case and prove their holomorphic property. We show that if the restrictions of maximal relations are continuously invertible, then the operators inverse to these restrictions are integral. By using these results, we prove the existence of the characteristic operator and describe the families of linear relations generating the characteristic operator. Определено семейства максимальных и минимальных отношений, порожденных интегральным уравнением с неванлинновской операторной мерой в бесконечномерном случае и доказано, что эти семейства голоморфны. Показано, что если сужения максимальных отношений непрерывно обратимы, то операторы, обратные к таким сужениям, являются интегральными. Используя эти результаты, доказываем существование характеристического оператора и описываем семейства линейных отношений, порождающих характеристический оператор. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the InЇnite-Dimensional Case Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the InЇnite-Dimensional Case |
| spellingShingle |
On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the InЇnite-Dimensional Case Bruk, V.M. |
| title_short |
On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the InЇnite-Dimensional Case |
| title_full |
On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the InЇnite-Dimensional Case |
| title_fullStr |
On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the InЇnite-Dimensional Case |
| title_full_unstemmed |
On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the InЇnite-Dimensional Case |
| title_sort |
on the characteristic operator of an integral equation with a nevanlinna measure in the inїnite-dimensional case |
| author |
Bruk, V.M. |
| author_facet |
Bruk, V.M. |
| publishDate |
2014 |
| language |
English |
| container_title |
Журнал математической физики, анализа, геометрии |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
We define the families of maximal and minimal relations generated by an integral equation with a Nevanlinna operator measure in the infinite-dimensional case and prove their holomorphic property. We show that if the restrictions of maximal relations are continuously invertible, then the operators inverse to these restrictions are integral. By using these results, we prove the existence of the characteristic operator and describe the families of linear relations generating the characteristic operator.
Определено семейства максимальных и минимальных отношений, порожденных интегральным уравнением с неванлинновской операторной мерой в бесконечномерном случае и доказано, что эти семейства голоморфны. Показано, что если сужения максимальных отношений непрерывно обратимы, то операторы, обратные к таким сужениям, являются интегральными. Используя эти результаты, доказываем существование характеристического оператора и описываем семейства линейных отношений, порождающих характеристический оператор.
|
| issn |
1812-9471 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/106790 |
| citation_txt |
On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the InЇnite-Dimensional Case / V.M. Bruk // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 2. — С. 163-188. — Бібліогр.: 24 назв. — англ. |
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AT brukvm onthecharacteristicoperatorofanintegralequationwithanevanlinnameasureintheinínitedimensionalcase |
| first_indexed |
2025-11-24T18:26:06Z |
| last_indexed |
2025-11-24T18:26:06Z |
| _version_ |
1850485953675657216 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2014, vol. 10, No. 2, pp. 163–188
On the Characteristic Operator of an Integral
Equation with a Nevanlinna Measure in the
Infinite-Dimensional Case
V.M. Bruk
Saratov State Technical University
77 Politekhnicheskaya Str., Saratov 410054, Russia
E-mail: vladislavbruk@mail.ru
Received January 19, 2013, revised August 20, 2013
We define the families of maximal and minimal relations generated by
an integral equation with a Nevanlinna operator measure in the infinite-di-
mensional case and prove their holomorphic property. We show that if the
restrictions of maximal relations are continuously invertible, then the ope-
rators inverse to these restrictions are integral. By using these results, we
prove the existence of the characteristic operator and describe the families
of linear relations generating the characteristic operator.
Key words: Hilbert space, linear relation, integral equation, characteris-
tic operator, Nevanlinna measure.
Mathematics Subject Classification 2010: 47A06, 47A10, 34B27.
1. Introduction
Integral equations with a Nevanlinna operator measure are sufficiently general
equations. For example, they include differential equations whose coefficients are
generalized functions [1], differential equations with holomorphic (with respect to
the spectral parameter) coefficients and with the Dirichlet integral whose imagi-
nary part is nonpositive [2], integro-differential equations with the Stieltjes inte-
grals [3] (see also references therein).
On a finite or infinite interval (a, b), we consider the integral equation
y(t) = y(t0)− iJ
t∫
t0
(dZ̃λ)y(s)− iJ
t∫
t0
(dṼ)f(s), (1)
The work is supported by the Russian Foundation of Basic Researches (grant 13-01-00378)
c© V.M. Bruk, 2014
V.M. Bruk
where y is the desired function. Here y, f are the functions with values belonging
to a separable Hilbert space H; J is an operator in H such that J∗ = J , J2 = E
(E is the identity operator); the function ∆→ Z̃λ(∆) is an operator measure on
(a, b) such that the measures ReZ̃i, Ṽ = (Imλ0)−1ImZ̃λ0 have locally bounded
variations on (a, b), ∆ is a Borel set; the function λ→ Z̃λ(∆) is the Nevanlinna
function for any fixed ∆, i.e., this function is holomorphic if Imλ 6= 0, Z̃∗λ(∆) =
Z̃λ̄(∆), and (Imλ)−1ImZ̃λ(∆) > 0.
Equation (1) generates holomorphic families of maximal and minimal rela-
tions. If the measure Z̃λ is absolutely continuous, then (1) can be reduced to a
differential equation with a Nevanlinna operator function. The linear relations
generated by this differential equation were studied in [4–6]. For the case where
H is a finite-dimensional space, equation (1) is considered in [7, 8]. The infinite-
dimensional case differs essentially from the finite-dimensional case. It can be
explained by the fact that the space H = L2(H, dṼ; a, b) is rather sophisticated
(H is the space in which the minimal and maximal relations are considered). The
elements of H are not necessarily the functions with values in H.
In the present paper, we define the families of maximal and minimal relations
generated by (1) in the infinite-dimensional case, study the properties of these re-
lations and describe the continuously invertible restrictions of maximal relations.
Thereby we generalize some assertions from [7, 8] to the infinite-dimensional case.
We apply the obtained results to prove the existence of the characteristic oper-
ator and to describe the families of linear relations generating the characteristic
operator.
We note that the definition of the characteristic operator for a differential
equation with a Nevanlinna operator function is given in [4, 5]. In these papers,
the existence of the characteristic operator is established by studying special
boundary value problems with a spectral parameter in the boundary condition.
2. Main Assumptions and Notations
Let H be a separable Hilbert space with the scalar product (·, ·) and the norm
‖·‖. Denote by B a set of bounded Borel subsets ∆ such that ∆ ⊂ (a, b). We
consider the function ∆→X̃ (∆) defined on B and ranging over a set of bounded
linear operators acting in H. The function X̃ is called the operator measure on
(a, b) (see, e.g., [9, ch. 5]) if X̃ is equal to zero on the empty set and the equality
X̃
( ∞⋃
n=1
∆n
)
=
∞∑
n=1
X̃ (∆n)
holds for disjoint Borel sets ∆n ∈ B, where the series converges weakly. By
164 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2
On the Characteristic Operator of an Integral Equation...
V∆(X̃ ), denote
V∆(X̃ ) = ρ̃(∆) = sup
∑
j
∥∥∥X̃ (∆j)
∥∥∥ ,
where ”sup” can be applied to the finite sums of disjoint Borel sets ∆j ⊂ ∆.
The number V∆(X̃ ) is called the variation of the measure X̃ on the Borel set ∆.
Suppose that the measure X̃ has a locally bounded variation, i.e., V[a1,b1](X̃ ) < ∞
for any segment [a1, b1] ⊂ (a, b). The function ∆→ ρ̃(∆) is a nonnegative measure
on (a, b). We assign ρ̃((a, b1]) = lim
n→∞ ρ̃((an, b1]) 6 ∞, where an → a as n→ ∞,
an > a, b1 < b (similarly for the endpoint b).
The following statement can be found in [9, ch. 5].
Statement 1. Suppose that the measure X̃ has a locally bounded variation
on (a, b). Then for ρ̃-almost all ξ ∈ (a, b) there exists an operator function ξ→
Ψ(ξ) such that Ψ has values in the set of bounded linear operators acting in H,
‖Ψ(ξ)‖ = 1, and the equality
X̃ (∆) =
∫
∆
Ψ(ξ)dρ̃ (2)
holds for all Borel sets ∆ ∈ B. The function Ψ is uniquely determined up to a
set of ρ-measure zero. In the case ρ̃(∆) < ∞, the integral sums for (2) converge
with respect to the usual norm of operators.
Let {∆, λ}→ Z̃λ(∆) be a function with values in the set of linear bounded
operators acting in H, where ∆ ∈ B, λ ∈ C0, C0 ⊃ C\R, the symbol {x1, x2}
denotes an ordered pair consisting of x1, x2. We assume that this function
is the Nevanlinna function for any fixed ∆, i.e., the following conditions hold:
(a) each point from C0 has a neighborhood (independent of ∆) such that the
function λ→ Z̃λ(∆) is holomorphic in this neighborhood; (b) Z̃∗λ(∆) = Z̃λ̄(∆);
(c) (Imλ)−1ImZ̃λ(∆) > 0 for all ∆ ∈ B and all λ such that Imλ 6= 0. Moreover,
this function satisfies condition (d). Before formulating condition (d), we intro-
duce some notations. We put Ṽλ(∆) = (Imλ)−1ImZ̃λ(∆). Then for all ν ∈ C0∩R
there exists (at least in the weak sense) lim
λ→ν±i0
Ṽλ(∆) = Ṽν(∆).
In [4], it was shown that conditions (a)–(c) imply
k1(Ṽλ(∆)g, g) 6 (Ṽµ(∆)g, g) 6 k2(Ṽλ(∆)g, g). (3)
It follows from [4, 6] that
∣∣∣(λ− µ)−1((Z̃λ(∆)− ReZ̃µ(∆))g, h)
∣∣∣ 6 k
∥∥∥Ṽ1/2
ζ (∆)g
∥∥∥
∥∥∥Ṽ1/2
η (∆)h
∥∥∥ . (4)
In inequalities (3), (4), g, h ∈ H, λ, µ, ζ, η ∈ C0, the constants k,k1,k2 > 0 are
independent of ∆∈B, λ, µ, ζ, η∈K (K is an arbitrary fixed compact, K⊂C0).
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 165
V.M. Bruk
Condition (d): the function ∆→ Z̃λ(∆) is an operator measure on (a, b) for
all λ ∈ C0, and the measures ReZ̃i, Ṽλ0 (for some point λ0 ∈ C0) have locally
bounded variations on (a, b).
It follows from condition (d) and (3), (4) that the measures Ṽλ, Z̃λ − ReZ̃µ,
Z̃λ (λ, µ ∈ C0) have locally bounded variations on (a, b). We fix some λ0 ∈ C and
put Ṽ = Ṽλ0 .
Suppose that the measure X̃ has a locally bounded variation on (a, b). We
call the endpoint a regular for the measure X̃ if a>−∞ and the measure X̃ has
a locally bounded variation on each segment [a, b1] (b1 < b). The endpoint a is
said to be singular if it is not regular. We define the regularity and singularity
of the endpoint b in a similar way.
In the case of regular endpoint, we extend the considered measures on a larger
interval as follows. We put a0 = a if the endpoint a is singular; if it is regular, we
fix a certain point a0 such that a0 < a, and we put X̃(∆) = 0 for all Borel sets
∆ ⊂ [a0, a). Analogously, we define a value b0 for the endpoint b. We also put
X̃(∆) = 0 for all Borel sets ∆ ⊂ (b, b0] in the case when the endpoint b is regular.
We use the previous notations for the extended measures. Note that equality (2)
remains valid for all Borel sets ∆ such that ∆ ⊂ (a0, b0).
It follows from (3), (4) that the endpoint a is regular for the measure Z̃λ if
and only if it is regular for the measures ReZ̃i and Ṽ = Ṽλ0 . The similar assertion
is valid for the endpoint b.
The integral
t∫
t0
is understood as the integral
∫
[t0,t)
if t > t0 or 0, if t = t0 or
−∫
[t,t0)
, if t < t0. In the integral
t∫
t0
, we assume that a0 < t0 < b0, a0 < t < b0. If
the endpoint a (or b) is regular, then it is possible that t0 =a0, t=a0 (or t0 =b0,
t=b0, respectively).
3. Solution of Integral Equations
A function f with values in H is called a step function on a Borel set ∆ if ∆
can be represented as the union of a finite number of disjoint Borel sets ∆j such
that f is constant on each ∆j . Obviously, the integral
∫
∆(dX̃ )f(t) exists for any
operator measure X̃ on (a, b) and for any step function f on ∆ ∈ B. Suppose the
measure X̃ has a locally bounded variation on (a, b), and ∆ is a Borel set with
the property ∆ ⊂ (a0, b0). Then the above integral is defined by the equality
∫
∆
(dX̃ )f(t) =
∫
∆
Ψ(t)f(t)dρ̃ (5)
166 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2
On the Characteristic Operator of an Integral Equation...
for a function f such that the Bochner integral exists in the right-hand side of
(5), where Ψ is the function from (2). Let {fn} be a sequence of step functions
on ∆. If
∫
∆ ‖f(s)− fn(s)‖ dρ̃ → 0 as n→∞, then
∫
∆
(dX̃ )f(t) = lim
n→∞
∫
∆
(dX̃ )fn(t).
The following integral is defined by the equality
∫
∆
((dX̃ )f(t), g(t)) =
∫
∆
(Ψ(t)f(t), g(t))dρ̃
under the condition that the integral exists in the right-hand side of this equality.
If ρ̃(∆)=∞, then the above integrals can be defined by the standard method.
It follows from (5) that if a Borel measurable function f is bounded and
ρ̃(∆) < ∞, then ∥∥∥∥∥∥
∫
∆
(dX̃ )f(t)
∥∥∥∥∥∥
6 sup
t∈∆
‖f(t)‖ ρ̃(∆). (6)
Let the function f be locally integrable with respect to the measure X̃ on
(a0, b0). We claim that the function y(t)=
∫ t
t0
(dX̃ )f(s) is continuous from the left
(here t0, t∈(a0, b0)). Indeed, if t<t1, then y(t1)−y(t)=
∫
[t,t1)(dX̃ )f(s). Using (2),
we get
‖y(t1)− y(t)‖ 6
∫
[t,t1)
‖Ψ(ξ)f(ξ)‖ dρ̃.
Now the desired assertion follows from the equality
⋂
t[t, t1) = ∅.
Suppose a segment [l1, l2] ⊂ (a0, b0). We consider a set of functions bounded
on [l1, l2], continuous from the left (in the strong sense) on (l1, l2], and ranging
over H. We introduce the norm ‖u‖[l1,l2] = sup
t∈[l1,l2]
‖u(t)‖ on this set and obtain
a Banach space denoted by C−[l1, l2].
Theorem 1. Suppose that the measure X̃ has a locally bounded variation on
(a, b), and the function h ∈ C−[a1, b1], where [a1, b1] ⊂ (a0, b0). Then for all
y0∈H there exists a unique solution of the equation
y(t) = y0 − iJ
t∫
t0
(dX̃ )y(s)− iJh(t) (a1 6 t0 6 b1, a1 6 t 6 b1) (7)
belonging to the space C−([a1, b1]).
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 167
V.M. Bruk
P r o o f. By the definition of the interval (a0, b0), the measure X̃ has the
bounded variation on the segment [a1, b1]. First, we will show that there exists a
segment Iδ = [t0 − δ, t0 + δ] such that equation (7) has a unique solution in the
space C−(Iδ) (δ > 0) for each y0 ∈ H. (If t0 = a1, we set Iδ = [t0, t0 + δ], and if
t0 = b1, we set Iδ = [t0 − δ, t0].)
Let t→ ρ(t) be a continuous from the left function generating the measure
ρ̃. By ρs(t0), we denote the jump of the function ρ at the point t0 (it is possible
that ρs(t0) = 0). We set r(t) = ρ(t) for t 6 t0 and r(t) = ρ(t)− ρs(t0) for t > t0.
The function r is continuous at t0. Let r̃ denote the measure generated by the
function r. We introduce the operator measure
Ỹ(∆) =
∫
∆
Ψ(ξ)dr̃. (8)
Under this notation, equation (7) has the form y = Ay + z, where
(Ay)(t)=−iJ
t∫
t0
(dỸ)y(s), z = y0 − iJX̃ ({t0})y0 − iJh(t). (9)
Taking into account (6), (8), (9), and the continuity of r, we obtain
‖(Ay)(t)‖ 6 sup
t∈Iδ
‖y(t)‖ |r(t)− r(t0)| < ε sup
t∈Iδ
‖y(t)‖ .
Consequently, sup
t∈Iδ
‖(Ay)(t)‖ 6 ε sup
t∈Iδ
‖y(t)‖. Using the continuity of r, we take
δ>0 under which ε < 1. Then ‖A‖C−(Iδ) <1. Hence the operator E −A has the
bounded everywhere defined inverse operator in the space C−(Iδ). Thus there
exists a unique solution of equation (7) on the interval Iδ. The solution is found
by using the formula y = (E−A)−1z. Now the desired statement can be obtained
in the standard way. The theorem is proved.
R e m a r k 1. If y is the solution of equation (7), then lim
t→t0−0
y(t) = y0.
Theorem 2. Assume that the operator measures X̃ , X̃n have locally bounded
variations on (a, b), [a1, b1] ⊂ (a0, b0), yn,0 ∈ H, hn ∈ C−[a1, b1], and assume that
the sequence {hn} converges uniformly to h on [a1, b1], and the sequence {yn,0}
converges to y0 in H. Let yn be the solution of the equation
yn(t) = yn,0 − iJ
t∫
t0
(dX̃n)yn(s)− iJhn(t), a1 6 t0 6 b1, a1 6 t 6 b1. (10)
If lim
n→∞V[a1,b1](X̃n − X̃ ) = 0, then the sequence {yn} converges uniformly to the
solution y of equation (7) on [a1, b1].
168 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2
On the Characteristic Operator of an Integral Equation...
P r o o f. It follows from the definition of the interval (a0, b0) that the mea-
sures X̃ , X̃n have bounded variations on the segment [a1, b1]. We construct the
measure Ỹn by the measure X̃n in the same way as the measure Ỹ is constructed
by the measure X̃ in the proof of Theorem 1. Then equation (10) can be written
as yn = Anyn + zn, where
(Anu)(t)=−iJ
t∫
t0
(dỸn)u(τ), zn = yn,0 − iJX̃n({t0})yn,0 − iJhn(t). (11)
We claim that the sequence {An} converges to A in the uniform operator
topology of the operators acting in C−(Iδ), where A, δ are the same as in the
proof of Theorem 1. Indeed, we denote ṽn = V[a1,b1](Ỹn−Ỹ). Then lim
n→∞ ṽn = 0.
From this, Statement 1, and inequality (6) applied to the measure Ỹn − Ỹ, we
obtain that for all function x ∈ C−(Iδ), for all ε > 0, the inequality
‖(An −A)x‖C−(Iδ) 6 sup
t∈Iδ
‖x(t)‖ ṽn(Iδ) < ε sup
t∈Iδ
‖x(t)‖ = ε ‖x‖C−(Iδ)
holds for large enough n.
Thus the sequence {An} converges to A in the uniform operator topology.
Therefore, for large enough n, the operator E −An has the bounded everywhere
defined inverse operator, and the sequence {(E−An)−1} converges to (E−A)−1
in the uniform operator topology of the operators acting in C−(Iδ). Using (9),
(11), we get y = (E − A)−1z, yn = (E − An)−1zn. Hence the sequence {yn}
converges to y in C−(Iδ) since {zn} converges to z in C−(Iδ). Now the desired
statement can be obtained in the standard way. The theorem is proved.
Lemma 1. Suppose the measures X̃λ satisfy the conditions: X̃λ have locally
bounded variations on (a, b); the function λ → X̃λ(∆) is holomorphic in some
neighborhood of the point λ1 for any fixed Borel set ∆ ⊂ [a1, b1] ⊂ (a0, b0) and
this neighborhood is independent of ∆; lim
λ→λ1
V[a1,b1](X̃λ−X̃λ1) = 0. If the function
t→ yλ(t) is the solution of equation (7), in which X̃ is changed to X̃λ, then the
point λ1 has a neighborhood independent of t ∈ [a1, b1] such that the function
λ→yλ(t) is holomorphic in this neighborhood for all t ∈ [a1, b1].
P r o o f. We construct the operator Aλ by the measure X̃λ in the same way
as the operator An is constructed by the measure X̃n in the proof of Theorem 2.
Then the function λ→Aλ is holomorphic in the neighborhood of λ1, and
lim
λ→λ1
(E −Aλ)−1 = (E −Aλ1)
−1.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 169
V.M. Bruk
Consequently, the equality lim
λ→λ1
yλ = yλ1 holds in the space C−(Iδ). Hence the
function λ→ yλ is holomorphic in the neighborhood of λ1 in the space C−(Iδ).
This implies the desired assertion. The lemma is proved.
Lemma 2. Let X̃1, X̃2, Ṽ be operator measures having locally bounded vari-
ations on (a, b), ranging over the set of bounded linear operators acting in H,
and Ṽ∗(∆) = Ṽ(∆) for all ∆ such that ∆ ⊂ (a0, b0); let the functions f , g be
locally integrable on (a0, b0) with respect to the measure Ṽ; y0, z0 ∈ H. Then for
all functions y, z, having the form
y(t)=y0 − iJ
t∫
t0
(dX̃1)y(s)− iJ
t∫
t0
(dṼ)f(s), z(t)=z0 − iJ
t∫
t0
(dX̃2)z(s)− iJ
t∫
t0
(dṼ)g(s),
the equality (the Lagrange formula)
c2∫
c1
((dṼ)f(t), z(t))−
c2∫
c1
(y(t), (dṼ)g(t)) = (iJy(c2), z(c2))− (iJy(c1), z(c1))
+
c2∫
c1
(y(t), (dX̃2)z(t))−
c2∫
c1
((dX̃1)y(t), z(t)) (12)
holds, where t0, t, c1, c2 ∈ (a0, b0).
The proof of the lemma is done by the routine transformations of the left-hand
side of equality (12). The transformations are carried out in the same way as in
[7], where the finite-dimensional case is considered. In these transformations, the
interchange of the order of integration is of great importance. It follows from (2)
that the interchange is possible in the infinite-dimensional case.
R e m a r k 2. In Lemma 2, if the endpoint a (or b) is regular, then for t,
t0, c1, c2 one can take a0 (or b0, respectively).
Let Wλ(t) denote the operator solution of the equation
Wλ(t)x0 = x0 − iJ
t∫
t0
(dZ̃λ)Wλ(s)x0 (x0∈H, λ∈C0, t0, t ∈ (a0, b0) ).
It follows from inequalities (3), (4) and Lemma 1 that the function λ→Wλ(t) is
holomorphic for all fixed t∈(a0, b0). The equality
W ∗̄
λ (t)JWλ(t) = J (13)
can be proved analogously to the corresponding equality in [7].
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On the Characteristic Operator of an Integral Equation...
Lemma 3. The function y is the solution of the equation
y(t) = x0 − iJ
t∫
t0
(dZ̃λ)y(s)− iJ
t∫
t0
(dṼ)f(s) (14)
if and only if y has the form
y(t) = Wλ(t)x0 −Wλ(t)iJ
t∫
t0
W ∗̄
λ (s)(dṼ)f(s), (15)
where f is locally integrable with respect to the measure Ṽ , x0 ∈ H, λ ∈ C0,
a0 < t0 < b0, a0 < t < b0.
P r o o f. It follows from Theorem 1 that there exists a unique solution
of equation (14). The limit as t→ t0−0 of the right-hand sides of (14), (15) is
equal to x0. Suffice it to prove that we will obtain the identity if we substitute
the right-hand side of equality (15) for y in (14). The substitution generates
transformations carried out in the same way as the corresponding transformations
in [7]. The lemma is proved.
We denote Ũ(∆) = ReZ̃i(∆) and consider the special case where Z̃λ(∆) =
Ũ(∆) + λṼ(∆) (λ ∈ C). Obviously, conditions (a)–(d) hold for this measure,
and ImZ̃λ(∆) = (Imλ)Ṽ(∆), ReZ̃i(∆) = Ũ(∆). Therefore all preceding results
remain valid for this case. In particular, Lemma 3 implies the following statement.
Corollary 1. The function y is the solution of the equation
y(t) = x0 − iJ
t∫
t0
(dŨ)y(s)− iλJ
t∫
t0
(dṼ)y(s)− iJ
t∫
t0
(dṼ)f(s) (16)
if and only if y has the form
y(t) = Wλ(t)x0 −Wλ(t)iJ
t∫
t0
W ∗̄
λ(s)(dṼ)f(s), (17)
where x0 ∈ H, λ ∈ C, t0, t ∈ (a0, b0), the function t →Wλ(t) is the operator
solution of the equation
Wλ(t)x0 = x0 − iJ
t∫
t0
(dŨ)Wλ(s)x0 − iλJ
t∫
t0
(dṼ)Wλ(s)x0.
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V.M. Bruk
4. The Space L2(H, dṼ; a, b) Generated by the Nevanlinna
Measure
On a set of step functions finite on the interval (a0, b0) and ranging over H,
we introduce the quasi-scalar product
(x, y)V =
b0∫
a0
((dṼ)x(t), y(t)). (18)
We identify the functions y such that (y, y)V = 0 with zero and perform the
completion. Then we obtain a Hilbert space denoted by H=L2(H, dṼ; a, b). The
elements of H are the classes of functions identical to the norm ‖y‖V =(y, y)1/2
V .
Here, to avoid complicating terminology, we denote the class of functions with
the representative y by the same symbol. We will also say that the function y
belongs to H. We treat the equalities between the functions belonging to H as
the equalities between the corresponding equivalence classes. The space H does
not change whenever we replace the interval (a0, b0) by (a′0, b
′
0), where the points
a′0, b′0 are introduced in the same way as the points a0, b0.
R e m a r k 3. It follows from (3) that the space H does not depend on
the choice of the point λ0∈C0 in the following sense. If we change the measure
Ṽ=Ṽλ0 to Ṽλ (λ ∈ C0) in (18), then we obtain the same set H supplied with an
equivalent norm.
We denote Z̃λ(∆) = Z̃λ(∆)− ReZ̃i(∆).
Lemma 4. The inequality
∣∣∣∣∣∣
b0∫
a0
((dZ̃λ)y(t), x(t))
∣∣∣∣∣∣
6 k ‖y‖V ‖x‖V (19)
holds for all functions y, x ∈ H, where k > 0 is independent of λ ∈ K (K is an
arbitrary fixed compact, K ⊂ C0).
P r o o f. Using (4), we obtain that inequality (19) holds for step functions.
Now the desired statement follows from the definition of the space H. The lemma
is proved.
It follows from Lemma 4 that the sesquilinear form in the left-hand side of
(19) is continuous on H × H. Hence there exists a bounded operator Zλ :H→H
such that
b0∫
a0
((dZ̃λ)y(t), x(t)) = (Zλy, x)V. (20)
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In (20), we take g = Zλy, x(t) = χ(t)x0, where x0 ∈H, χ(t) is the characteristic
function of the Borel set ∆. Then we obtain
∫
∆((dZ̃λ)y(t), x0)=
∫
∆((dṼ)g(t), x0).
Consequently, ∫
∆
(dZ̃λ)y(t) =
∫
∆
(dṼ)(Zλy)(t) (21)
for any Borel set ∆ and any function y ∈ H.
Theorem 3. The operator function λ→Zλ is holomorphic on C0.
P r o o f. Taking into account condition (a) and equality (20), we obtain
that the function λ → Zλy is holomorphic for any finite step-function y. Now
the desired statement follows from Lemma 4 and the density of the set of finite
step-functions in H. The theorem is proved.
We study the structure of the space H in detail. It follows from condition (d)
that there exists an operator function Ψ satisfying all conditions of Statement 1
in which the measure X̃ is changed to the measure Ṽ. In particular, the equality
Ṽ(∆) =
∫
∆ Ψ(ξ)dρ̃ holds.
The inequality Ṽ(∆) > 0 implies Ψ(ξ) > 0 for ρ-almost all ξ ∈ (a0, b0). We use
some constructions from [10]. We denote G(t) = kerΨ(t); H(t) = HªG(t); Ψ0(t)
is the restriction of Ψ(t) to H(t). Then the operator Ψ0(t) acting in H(t) has the
inverse Ψ−1
0 (t) (which, in general, is unbounded). Let {Hτ (t)} (−∞ < τ < ∞) be
a Hilbert scale of spaces generated by the operator Ψ−1
0 (t) [9, ch. 3], [11, ch. 2].
It follows from the definition of the Hilbert scale that the operator Ψ0(t) can be
extended to the operator Ψ̂0(t), which maps H−α(t) continuously and bijectively
onto H1−α(t) (α > 0). Let Ψ̂(t) denote the operator defined on H−α(t) ⊕ G(t)
and coinciding with Ψ̂0(t) on H−α(t) and with zero on G(t). The operator Ψ̂(t) is
the extension of Ψ(t). Below the case α = 1/2 is considered. Thus, the operator
Ψ̂(t) maps H−1/2(t)⊕G(t) continuously onto H1/2(t).
In [10], the case, where the measure ρ̃ is the usual Lebesgue measure, i.e.,
ρ̃([a1, b1)) = b1−a1, is considered. By literally repeating argumentation from [10],
it is proved that the spaces H−1/2(t) are ρ̃-measurable with respect to the pa-
rameter t [12, ch. 1] whenever for measurable functions the functions of the form
t → Ψ̂−1
0 (t)Ψ1/2(t)h(t) are taken, where h is an arbitrary ρ̃-measurable func-
tion ranging over H. The space H is a measurable sum of the spaces H−1/2(t)
with respect to the measure ρ̃, and H consists of all functions of the form
t → Ψ̂−1
0 (t)Ψ1/2(t)g(t), where g is an arbitrary ρ̃-measurable function ranging
over H such that
∫ b0
a0
‖g(t)‖2 dρ̃ < ∞. We note that the above description of the
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V.M. Bruk
space H follows also from [13]. Thus we obtain that the equality
(x, y)V =
b0∫
a0
(Ψ̂(t)x(t), y(t))dρ̃ =
b0∫
a0
(Ψ1/2(t)x(t), Ψ1/2(t)y(t))dρ̃ (22)
holds for all functions x, y ∈ H. It follows from (2), (22) that
∫
∆(dṼ)y(t) ∈ H for
all functions y ∈ H and for all Borel sets ∆ with the property ∆ ⊂ (a0, b0).
5. Maximal and Minimal Relations in the Regular Case
Let B1, B2 be Banach spaces. A linear relation T is understood as a linear
manifold T ⊂ B1×B2. The terminology of the linear relations can be found, for
example, in [14]. From now onwards, the following notations are used: kerT is a
set of the elements x ∈ B1 such that {x, 0} ∈ T; KerT is a set of ordered pairs of
the form {x, 0} ∈ T; D(T) is a domain of T; R(T) is a range of T. The relation T
is called continuously invertible if T−1 is a bounded everywhere defined operator.
Linear operators are treated as linear relations. Since all relations considered
further are linear, the word ”linear” will often be omitted.
A family of linear relations is understood as a function λ→T(λ) (λ ∈ D ⊂ C),
where T(λ) is a linear relation, T(λ) ⊂ B1 × B2. A family of closed relations
T(λ) is called holomorphic at a point λ1 ∈ C if there exists a Banach space B0
and a family of bounded linear operators K(λ) : B0 → B1 × B2 such that the
operator K(λ) maps B0 bijectively onto T(λ) for any fixed λ, and the family
λ → K(λ) is holomorphic in some neighborhood of λ1. A family of relations is
called holomorphic on the domain D if it is holomorphic at all points belonging
to D. These definitions generalize the corresponding definitions of holomorphic
families of closed operators [15, ch. 7].
Now we introduce the linear relations generated by equations (14), (16) for
the case of regular endpoints. We represent integral equation (14) as
y(t) = x0 − iJ
t∫
t0
(dŨ)y(s)− iJ
t∫
t0
(dZ̃λ)y(s)− iJ
t∫
t0
(dṼ)f(s). (23)
Let L′(λ) (L′, respectively) be the relation consisting of the pairs {ỹ, f̃}∈H×H
satisfying the condition: for each pair {ỹ, f̃} there exists a pair {y, f} such that
the pairs {ỹ, f̃}, {y, f} are identical in H × H, and equality (14) (equality (16)
for λ = 0, respectively) holds on (a0, b0). By L(λ) (by L), denote the closure
of L′(λ) (of L′) and call L(λ) (L) the maximal relation generated by integral
equation (14) (equation (16) for λ = 0, respectively). We define the minimal
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On the Characteristic Operator of an Integral Equation...
relation L0(λ) (L0) as the restriction of L′(λ) (L′) to the set of functions y such
that y(a0)=y(b0)=0, where y is a solution of (14) ((16) for λ = 0, respectively).
R e m a r k 4. It follows from the definition of the points a0, b0 that
y(a0)= lim
t→a−0
y(t), y(b0)= lim
t→b+0
y(t). The maximal and minimal relations do not
change if we replace the interval (a0, b0) by (a′0, b
′
0), where the points a′0, b′0 are
defined in the same way as a0, b0. Thus the minimal relations L0(λ), L0 can be
defined as the restrictions of L′(λ), L′ to a set of the functions y finite on (a0, b0),
where y is a solution of (14) or (16) for λ=0, respectively. We note that in [7], [8]
the minimal relations should be defined in the same way, i.e., as the restrictions
of L′(λ), L′ to a set of the functions y finite on (a0, b0) (in the finite-dimensional
case, L′(λ)=L(λ), L′=L).
Using (21) and (23), we obtain
L(λ) = L − Zλ, L0(λ) = L0 − Zλ. (24)
Let Q0 be a set of the elements x ∈ H such that the function t → Wµ(t)x
(µ∈C0) is identical to zero in H. We denote Q = H ªQ0. We claim that the set
Q0 (and therefore Q) does not change if we substitute Wλ (λ∈C0) for Wµ or Wλ
(λ∈C) for Wµ. Indeed, let uµ =Zµ(Wµ(·)x), vµ =Zλ(Wµ(·)x), wλ =Zµ(Wλ(·)x).
Using Lemma 3, Corollary 1 and equality (21), we get
Wµ(t)x = Wλ(t)x−Wλ(t)iJ
t∫
t0
W ∗̄
λ (s)(dṼ)(uµ(s)− vµ(s)), (25)
Wµ(t)x = Wλ(t)x−Wλ(t)iJ
t∫
t0
W ∗̄
λ(s)(dṼ)(uµ(s)− λWµ(s)x), (26)
Wλ(t)x = Wµ(t)x−Wµ(t)iJ
t∫
t0
W ∗
µ̄(s)(dṼ)(λWλ(s)x− wλ(s)) (27)
for all x ∈ H. It follows from (25) that Wλ(t)x is identical to zero in H if x ∈ Q0.
Substituting λ for µ and µ for λ in (25), we obtain the converse assertion. By
(26), (27), it follows that Q0 does not change if we substitute Wλ for Wµ.
In Q, we introduce the norm
‖c‖− =
b0∫
a0
((dṼ)Wµ(s)c,Wµ(s)c)
1/2
=
b0∫
a0
∥∥∥Ψ1/2(s)Wµ(s)c
∥∥∥
2
dρ̃
1/2
6 γ ‖c‖ , µ ∈ C0, c ∈ Q, γ > 0. (28)
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V.M. Bruk
We denote the completion of Q with respect to this norm by Q−. It follows from
(25)–(27) that the replacement of Wµ(s) by Wλ(s) (λ ∈ C0) or of Wµ(s) by Wλ
(λ ∈ C) in (28) leads to the same set Q− with the equivalent norm. The space
Q− can be treated as a space with negative norm with respect to Q [9, ch. 1],
[11, ch. 2]. By Q+, denote the corresponding space with positive norm.
Suppose the sequence {cn} (cn ∈Q) converges to c0 ∈Q− in Q−. Then the
sequences {Wλ(·)cn}, {Wλ(·)cn} are fundamental in H and hence converge in H
to some elements from H. Denote these elements by Wλ(·)c0 and Wλ(·)c0. Let
W(λ) and w(λ) denote the operators c→Wλ(·)c and c→Wλ(·)c, respectively,
where c∈Q−. The operators W(λ), w(λ) are continuous one-to-one mappings
of Q− into H and their ranges are closed. Hence the adjoint operators W∗(λ),
w∗(λ) map H continuously onto Q+. For all x∈Q, f ∈H, we have
(f,W(λ)x)V =
b0∫
a0
((dṼ)f(s),Wλ(s)x)=
b0∫
a0
(W ∗
λ (s)(dṼ)f(s), x) = (W∗(λ)f, x).
The analogous equality holds for the operator w(λ). Hence, taking into account
that Q is densely embedded in Q−, we obtain
W∗(λ)f =
b0∫
a0
W ∗
λ (s)(dṼ)f(s), w∗(λ)f =
b0∫
a0
W∗
λ(s)(dṼ)f(s). (29)
Thus the following statement is obtained.
Lemma 5. The operators W∗(λ), w∗(λ) map H continuously onto Q+ and
have the form (29).
The following lemma and corollaries can be proved in the same way as the
corresponding assertions in [6–8].
Lemma 6. The pair {ỹ, f̃} ∈ H×H belongs to the relation L(λ) (the relation
L− λE) if and only if there exists a pair {y, f} such that the pairs {ỹ, f̃}, {y, f}
are identical in H×H, and equality (15) (equality (17), respectively) holds, where
x0 ∈ Q−, f ∈ H.
Corollary 2. The relations L0(λ), L0 are closed.
Corollary 3. The range of the relation L0(λ) (relation L0 − λE) consists of
all elements f ∈ H such that the following equalities hold, respectively:
W∗(λ)f =
b0∫
a0
W ∗̄
λ (s)(dṼ)f(s) = 0, w∗(λ)f =
b0∫
a0
W ∗̄
λ(s)(dṼ)f(s) = 0.
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Corollary 4. The operators W(λ) and w(λ) are continuous one-to-one map-
pings of Q− onto kerL(λ) and onto ker(L − λE), respectively.
Theorem 3, Corollary 2 and equalities (24) imply the following statement.
Corollary 5. The families of the relations L(λ),L0(λ)are holomorphic on C0.
Lemma 7. L0 is a symmetric relation and L = L∗0.
P r o o f. In Lemma 2, we take X̃1 = X̃2 = Ũ , Ṽ = Ṽ, c1 = a0, c2 = b0. It
follows from (12) that L0 is the symmetric relation and L′ ⊂ L∗0. Consequently,
L ⊂ L∗0. We prove the converse inclusion. Using Corollaries 3, 4, we get Nλ =
ker(L − λ̄E), where Nλ is the defect subspace of the relation L0, i.e., Nλ is the
orthogonal complement of the range R(L0−λE). By N̂λ, denote a set of all pairs
of the form {z, λ̄z}, where z ∈ Nλ (Imλ 6= 0). It follows from Lemma 6 and the
equality L∗0 = L0 u N̂λ u N̂λ̄ that L∗0 ⊂ L. The lemma is proved.
Corollary 6. The equation L∗0(λ) = L(λ̄) (λ ∈ C0) holds.
Theorem 4. The equality
Im(f, y)V = −
b0∫
a0
((dImZ̃λ)y(t), y(t)) (30)
holds for all λ ∈ C0 and for all pairs {y, f} ∈ L0(λ).
P r o o f. In Lemma 2, we take X̃1 = X̃2 = Z̃λ, Ṽ = Ṽ, z = y, g = f , c1 = a0,
c2 =b0. The desired statement follows from (12) and the definition of L0(λ).
Corollary 7. The inequality (Imλ)−1Im(f, y)V 6−k ‖y‖2
V holds for any fixed
λ (Imλ 6= 0) and for all pairs {y, f} ∈ L0(λ), where k > 0.
We construct a space of boundary values for the relation L(λ). According to
Lemma 6, the pair {ỹ, f̃} ∈ H×H belongs to the relation L(λ) if and only if there
exists a pair {y, f} such that the pairs {ỹ, f̃}, {y, f} are identical in H× H, and
the equality
y(t) = Wλ(t)cλ + Fλ(t) (31)
holds, where cλ ∈ Q−,
Fλ(t) = −Wλ(t)iJ
t∫
a0
W ∗̄
λ (s)(dṼ)f(s)ds. (32)
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V.M. Bruk
To each pair {y, f} ∈ L(λ), we assign a pair of boundary values {yλ,y′λ} ∈
Q−×Q+ by the formulas
y′λ = W∗(λ)f =
b0∫
a0
W ∗̄
λ (s)(dṼ)f(s) ∈ Q+, yλ = cλ − (1/2)iJy′λ ∈ Q−.
It follows from (31), (32) that if the pairs {ỹ, f̃}, {y, f} are identical in H×H, then
their boundary values coincide. Let γ(λ), γ1(λ), γ2(λ) be the operators defined
by the equalities γ(λ){y, f} = {yλ,y′λ}, γ1(λ){y, f} = yλ, γ2(λ){y, f} = y′λ. It
follows from Lemma 5 and Corollary 4 that γ(λ) maps L(λ) continuously onto
Q−× Q+, and the restriction of γ1(λ) to KerL(λ) is the one-to-one mapping
of KerL(λ) onto Q−. Hence the quadruple (Q−, Q+, γ1(λ), γ2(λ)) is the space
of boundary values for the relation L(λ) from the viewpoint of [16] (see also
references in [6]).
For fixed λ∈C0, between the relations L̂(λ) with the property L0(λ)⊂ L̂(λ)⊂
L(λ) and the relations θ(λ)⊂Q−×Q+ there exists a one-to-one correspondence
determined by the equality γ(λ)L̂(λ) = θ(λ). In this case we denote L̂(λ) =
Lθ(λ)(λ).
Lemma 8. For all pairs {y, f} ∈ L(λ), {z, g} ∈ L(λ̄) (λ ∈ C0), ”the Green
formula”
(f, z)V − (y, g)V = (y′λ, zλ̄)− (yλ, z′̄λ) (33)
holds, where γ(λ){y, f} = {yλ,y′λ}, γ(λ){z, g} = {zλ, z′λ}.
P r o o f. According to Lemma 6, the function y has the form (31). Anal-
ogously, the function z can be represented as z(t) = Wλ̄(t)dλ̄ +Gλ̄(t), where
dλ̄ ∈ Q−, Gλ̄(t) = −Wλ̄(t)iJ
∫ t
a0
W ∗
λ (s)(dṼ)g(s)ds. In (12), we take X̃1 = Z̃λ,
X̃2 = Z̃λ̄, z(t)=Gλ̄(t), y(t)=Fλ(t), c1 =a0, c2 =b0. Using (13) and Lemmas 2, 3,
we get
(f, Gλ̄)V − (Fλ, g)V = (iJWλ(b0)iJy′λ,Wλ̄(b0)iJzλ̄) = (iJy′λ, z′̄λ). (34)
We take two sequences {cλ,n}, {dλ̄,n} such that cλ,n, dλ̄,n ∈ Q and {cλ,n}, {dλ̄,n}
converge to cλ, dλ̄ ∈ Q− in Q−, respectively. We denote vn(t) = Wλ(t)cλ,n,
un(t) = Wλ̄(t)dλ̄,n, v(t) = Wλ(t)cλ, u(t) = Wλ̄(t)dλ̄. Then the sequences {vn},
{un} converge to v, u in H, respectively. From (13) and Lemmas 2, 3, we obtain
(f, un)V = (f, un)V − (Fλ, 0)V = −(iJWλ(b0)iJy′λ,Wλ̄(b0)dλ̄,n) = (y′λ, dλ̄,n),
−(vn, g)V = (0, Gλ̄)V − (vn, g)V = −(iJWλ(b0)cλ,n,Wλ̄(b0)iJz′̄λ) = −(cλ,n, z′̄λ).
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In these equalities, we pass to the limit as n→∞. Then we get
(f, u)V = (y′λ, dλ̄), (v, g)V = (cλ, z′̄λ). (35)
Using (34), (35), we obtain
(f, z)V − (y, g)V = (y′λ, dλ̄)− (cλ, z′̄λ) + (iJy′λ, z′̄λ) =
= (y′λ, dλ̄ − (1/2)iJz′̄λ)− (cλ − (1/2)iJy′λ, z′̄λ) = (y′λ, zλ̄)− (yλ, z′̄λ).
The lemma is proved.
Corollary 8. The equation (Lθ(λ)(λ))∗ = Lθ∗(λ)(λ̄) holds.
In view of Lemma 6 and Corollary 8, the following theorem can be proved
analogously to the corresponding assertions in [6, 8].
Theorem 5. For any fixed λ ∈ C0, the relation L̂(λ) = Lθ(λ)(λ) is contin-
uously invertible if and only if the relation θ(λ) has the same property. In this
case, the operator R(λ)= L̂−1(λ) is integral,
R(λ)g =
b0∫
a0
K(t, s, λ)(dṼ)g(s) (g ∈ H),
where K(t, s, λ)=Wλ(t)(θ−1(λ)+(1/2)sgn(s−t)iJ)W ∗̄
λ
(s). The equalities θ∗(λ)=
θ(λ̄) and R∗(λ) = R(λ̄) hold simultaneously. The function λ→Rλ (λ ∈ D ⊂ C0)
is holomorphic at a point λ1 ∈ D if and only if the function λ → θ−1(λ) is
holomorphic at the same point.
6. Maximal and Minimal Relations in the Singular Case
Let L′0(λ) (L′0, respectively) be the relation consisting of the pairs {ỹ, f̃} ∈
H×H satisfying the conditions: for each pair {ỹ, f̃} there exists a pair {y, f}
such that the pairs {ỹ, f̃}, {y, f} are identical in H × H; the function y is finite
on (a0, b0); and equality (14) (equality (16) for λ = 0, respectively) holds. We
define the minimal relations L0(λ), L0 as the closures of the relations L′0(λ), L0,
respectively. The relations (L0(λ̄))∗=L∗0(λ̄), L∗0 are maximal relations.
Using (21), (23), we get L′0(λ) = L′0 − Zλ. Consequently, L0(λ) = L0 − Zλ.
It follows from Theorem 3 that the family λ → L0(λ) is holomorphic on C0.
Therefore λ→ L∗0(λ̄) is the holomorphic family (see [6]). Thus, the families of
minimal and maximal relations are holomorphic on C0. Suppose {y, f} ∈ L′0(λ).
In Lemma 2, we take X̃1 = X̃2 = Z̃λ, Ṽ = Ṽ, z = y, g = f and choose c1, c2 such
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 179
V.M. Bruk
that y(c1)=y(c2)=0. Then equality (30) holds. Passing to the limit in (30), we
can see that Theorem 4 remains valid for L0(λ) in the singular case.
Let Q0 be a set of the elements c ∈ H such that
b0∫
a0
((dṼ)Wµ(s)c,Wµ(s)c) = 0.
We denote Q = HªQ0. The sets Q0, Q do not depend on µ ∈ C0. (This assertion
can be proved analogously to the corresponding assertion for the regular case.)
Let [αn, βn] be a sequence of intervals such that a0 <αn+1 <αn <βn <βn+1 <b0
and αn→a0, βn→b0 as n→∞. In Q, we introduce a system of semi-norms
pn(x) =
∫
[αn,βn]
((dṼ)Wµ(s)c,Wµ(s)c)
1/2
, µ ∈ C0, c ∈ Q. (36)
By Q−, denote the completion of Q with respect to this system. The space Q− is
generated by the counting system of semi-norms. Hence Q− is a Frechet space [17,
ch. 2]. Arguing as above, we can see that the replacement of µ by λ ∈ C0 in (36)
leads to an isomorphic space.
We define the function t→Wλ(t)c for all c ∈ Q−. The reasonings are similar
to those given in [18, 6] and thus are omitted.
By Hn, denote Hn = L2(H, dṼ; [αn, βn]). Let Q0(n) be a set of the elements
c ∈ Q such that the function t → Wλ(t)c is identical to zero in the space Hn,
Q(n) = Q ª Q0(n). Obviously, Q(n) ⊃ Q(m) for n > m. If c ∈ Q(n), then
pn(c) > 0. Hence pn is the norm on the set Q(n). We denote it by ‖·‖(n)
− . Let
Q−(n) be the completion of Q(n) with respect to the norm ‖·‖(n)
− .
We define the mappings hmn : Q−(n) → Q−(m) (n > m) as follows. Let
Q(n,m) = Q(n) ª Q(m). For all elements c1 ∈ Q(m), c0 ∈ Q(n,m), we put
hmnc1 = c1, hmnc0 =0. Using the equality pm(c0) = 0, we obtain
‖hmn(c1 + c0)‖(m)
− = ‖c1‖(m)
− = pm(c1) = pm(c1 + c0) 6 ‖c1 + c0‖(n)
− .
This implies that the mapping hmn is extended by the continuity to the space
Q−(n).
We consider the projective limit lim(pr)hmnQ−(n) of the family of the spaces
{Q−(n);n ∈ N} with respect to the mappings hmn (m,n ∈ N,m 6 n) [17, ch. 2].
Repeating the corresponding arguments from [17, ch. 2, proof 5.4], one can show
that this projective limit is isomorphic to the space Q− introduced after formula
(36). It follows from the definition of projective limit that Q− is a subspace
180 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2
On the Characteristic Operator of an Integral Equation...
of the product
∏
n Q−(n), and Q− consists of all elements c = {cn} such that
cm = hmncn for all m, n, where m 6 n.
The space Q−(n) can be treated as a negative one with respect to Q(n) [9,
ch. 1], [11, ch. 2]. By Q+(n), denote the corresponding space with positive norm.
Then Q+(m) ⊂ Q+(n) for m 6 n and the inclusion map of Q+(m) into Q+(n)
is continuous. This inclusion map coincides with h+
mn, where h+
mn : Q+(m) →
Q+(n) is the adjoint mapping of hmn. By Q+, denote the inductive limit [17,
ch. 2] of the family {Q+(n);n ∈ N} with respect to the mappings h+
mn, i.e.,
Q+ = lim(ind)h+
mnQ+(n). According to [17, ch. 4], Q+ is the adjoint space of
Q−. The space Q+ can be treated as the union Q+ =∪nQ+(n) with the strongest
topology such that all inclusion maps of Q+(n) into Q+ are continuous [17, ch.
2].
By Corollary 4, the operator cn→Wλ(t)cn is a continuous one-to-one mapping
of Q−(n) into Hn and it has the closed range. We denote this operator by Wn(λ).
It follows from Lemma 5 that the adjoint operator W∗
n(λ) maps Hn continuously
onto Q+(n), and
W∗
n(λ)f =
βn∫
αn
W ∗
λ (s)(dṼ)f(s).
Consequently,
β∫
α
W ∗
λ (s)(dṼ)f(s) ∈ Q+
for each function f ∈ H and for all α, β such that a0 < α < β < b0.
Suppose c = {cn} ∈ Q−. Then cm = hmncn (m 6 n). Hence the restriction of
the function Ψ̂1/2(t)Wλ(t)cn to the segment [αm, βm] coincides with the function
Ψ̂1/2(t)Wλ(t)cm in the space L2(H, dρ̃; [αm, βm]). By Ψ̂1/2(t)Wλ(t)c, we denote
the function equal to Ψ̂1/2(t)Wλ(t)cn on each segment [αn, βn]. Now by Wλ(t)c
denote the function ranging over H−1/2(t) ⊕ G(t) and coinciding with Wλ(t)cn
in the space Hn for any n ∈ N. For all m, n (m 6 n), the equality Wλ(t)cn =
Wλ(t)cm holds in the space Hm.
From Lemma 6, we obtain the following statement.
Lemma 9. If the pair {ỹ, f̃} ∈ L∗0(λ̄), then there exists a pair {y, f} such that
the pairs {ỹ, f̃}, {y, f} are identical in H× H, and the equality
y(t) = Wλ(t)c−Wλ(t)iJ
t∫
t0
W ∗̄
λ (s)(dṼ)f(s)ds, c ∈ Q−, t0, t ∈ (a0, b0)
holds.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 181
V.M. Bruk
The following theorem can be proved in the similar way as the corresponding
assertions in [6, 8, 18]. Note that in [6, 18], the linear relations satisfy the re-
quirements under which the boundedness condition for the function λ→‖R(λ)‖
on some neighborhood of the point λ1 holds automatically.
Theorem 6. Assume that the relation L̂(λ) has the property L0(λ)⊂ L̂(λ)⊂
L∗0(λ̄). If the relation L̂(λ) is continuously invertible, then the operator R(λ) =
L̂−1(λ) is integral,
R(λ)g =
b0∫
a0
K(t, s, λ)(dṼ)g(s)ds, g ∈ H, (37)
where K(t, s, λ) = Wλ(t)(M(λ) + (1/2)sgn(s − t)iJ)W ∗̄
λ
(s), M(λ) : Q+ → Q−
is the continuous operator. The equalities R∗(λ) = R(λ̄), M∗(λ) = M(λ̄) hold
simultaneously. The integral (37) converges at least weakly in the space H. If
the function λ→R(λ) (λ ∈D ⊂C0) is holomorphic at a point λ1 ∈D, then the
function λ→M(λ)x is holomorphic at the same point for all x ∈ Q+. The con-
verse statement is valid whenever the function λ→‖R(λ)‖ is bounded on some
neighborhood of the point λ1.
7. The Characteristic Operator
In this section, the endpoints of the interval (a, b) may be regular or singular.
In the space H = L2(H, dṼ; a, b), the scalar product is defined by equality (18).
According to Remark 3, the replacement of Ṽ=Ṽλ0 by Ṽλ (λ∈C0) in (18) leads to
the same set H with an equivalent norm (the designation of Ṽλ is given before for-
mula (3)). By ‖·‖Vλ
(by (·, ·)Vλ
), we denote the norm (the scalar product, respec-
tively) in the space L2(H, dṼλ; a, b). According to these notations, ‖·‖H =‖·‖V.
We replace Ṽ by Ṽλ (λ ∈ C0) in (14) (or in (1)). Then we obtain the equation
y(t) = y(t0)− iJ
t∫
t0
(dZ̃λ)y(s)− iJ
t∫
t0
(dṼλ)f(s). (38)
The following definition of the characteristic operator for a differential equation
with a Nevanlinna operator function is given in [4, 5].
Definition 1. Let λ→M(λ)=M∗(λ̄) be a function holomorphic for Imλ 6=0
whose values are bounded linear operators and D(M(λ)) = Q+, R(M(λ))⊂ Q−.
This function M is called a characteristic operator of equation (38) if the operator
182 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2
On the Characteristic Operator of an Integral Equation...
R(λ), defined by the equality
(R(λ)f)(t)=
b0∫
a0
Wλ(t)(M(λ) + (1/2)sgn(s− t)iJ)W ∗̄
λ (s)(dṼλ)f(s), (39)
satisfies the inequality
‖R(λ)f‖2
Vλ
6 Im(R(λ)f, f)Vλ
/Imλ, Imλ 6= 0 (40)
for all function f ∈ H such that f is finite on (a0, b0).
Using (40), we get the inequality ‖R(λ)f‖Vλ
6 (Imλ)−1 ‖f‖Vλ
(Imλ 6= 0) for
all finite functions f ∈ H. Hence the operator R(λ) is bounded in H. Therefore
R(λ) is extended by the continuity to the whole space H. We denote the extended
operator by the same symbol R(λ).
We claim that for any function f ∈ H and for any fixed λ ∈ C0 there exists a
unique function g ∈ H such that the equality
t∫
t0
(dṼλ)f(s) =
t∫
t0
(dṼ)g(s) (41)
holds for all t ∈ (a0, b0). Indeed, we consider the sesquilinear form (f, u)Vλ
, where
u is an arbitrary function belonging to H. This form is continuous on H × H.
Consequently, there exists a unique function g ∈ H such that (f, u)Vλ
=(g, u)V.
From this equality, we obtain the desired assertion.
We consider the operator R(λ) defined by the equality
(R(λ)g)(t)=
b0∫
a0
Wλ(t)(M(λ) + (1/2)sgn(s− t)iJ)W ∗̄
λ (s)(dṼ)g(s), (42)
where g ∈ H, and g is finite on (a0, b0). Suppose that the functions f, g ∈ H are
connected by equality (41). Using (39), (42), we get R(λ)g =R(λ)f . Therefore
inequality (40) is equivalent to the following inequality:
‖R(λ)g‖2
Vλ
6 (Imλ)−1Im(R(λ)g, g)V, Imλ 6= 0. (43)
Then, since the norms ‖·‖V and ‖·‖Vλ
are equivalent, the operator R(λ) is
bounded in the space H. Consequently R(λ) is extended by the continuity to
the whole space H. We denote the extended operator by the same symbol R(λ).
It follows from (43) that the function λ → ‖R(λ)‖ is bounded in some neigh-
borhood of the arbitrary point λ1 (Imλ1 6= 0). Theorems 5, 6 imply that the
functions λ→R(λ) and λ→M(λ) are simultaneously holomorphic for Imλ 6= 0.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 183
V.M. Bruk
Suppose that a family of closed relations λ → L(λ) satisfies the condition
L0(λ)⊂L(λ)⊂L∗0(λ̄) for Imλ 6=0. We say that this family generates a character-
istic operator M(λ) if the equality L−1(λ) = R(λ) holds.
Now we need the description of the families generating the characteristic ope-
rator. Let H1, H2 be Hilbert spaces. By F(H1,H2), denote a set of holomorphic
functions λ→F (λ) (Imλ 6= 0) whose values are bounded operators satisfying the
conditions: (i) D(F (λ)) = H1, R(F (λ)) ⊂ H2 for Imλ > 0; (ii) F (λ̄) = F ∗(λ);
(iii) ‖F (λ)‖61. Let Ni = HªR(L0− iE) = ker(L∗0 + iE) be the defect subspace
of the relation L0. We consider the holomorphic operator function λ→F (λ) be-
longing to F(Ni, N−i). By LF (λ), denote the linear relation consisting of all pairs
of the form {y0 + F (λ)z0 − z0, y1 + iF (λ)z0 + iz0}, where {y0, y1} ∈ L0, z0 ∈ Ni.
Then (LF (λ))∗ =LF (λ̄). The family λ→LF (λ) is holomorphic for Imλ 6= 0. The
relations LF (λ) are maximal accumulative for Imλ > 0 and maximal dissipative
for Imλ<0. Conversely, let λ→L̂(λ) (Imλ 6=0) be a holomorphic family of linear
relations with the properties: L0⊂ L̂(λ)⊂L∗0; L̂∗(λ) = L̂(λ̄); each relation L̂(λ)
is maximal accumulative for Imλ > 0. Then there exists an operator function
λ→F (λ) belonging to F(Ni,N−i), and L̂(λ)=LF (λ) (see [19, 20]).
Theorem 7. If the function λ→F (λ) belongs to F(Ni,N−i), then the family
of relations L(λ) = LF (λ)− Zλ generates a characteristic operator of equation
(38) and, conversely, if the family λ→L(λ) (Imλ 6=0) generates a characteristic
operator, then there exists an operator function λ→F (λ) belonging to F(Ni, N−i)
such that L(λ) = LF (λ) − Zλ.
P r o o f. First notice that the inclusions L0(λ)⊂ L(λ)⊂ L∗0(λ̄) and L0 ⊂
LF (λ) ⊂ L∗0 hold simultaneously. It follows from Theorem 3 that the family
λ→ L(λ) is holomorphic if and only if the family λ→LF (λ) is the same. The
equalities L(λ̄) = L∗(λ) and (LF (λ))∗ = LF (λ̄) hold simultaneously since Z∗λ = Zλ̄.
Using (20), we get
Im(Zλf0, f0)V = (Imλ)
b0∫
a0
((dṼλ)f0(t), f0(t))
for all functions f0 ∈ H. Therefore, if L(λ) is continuously invertible, then
inequality
(Imλ)−1Im(g, z)V 6 −(Imλ)−1Im(Zλz, z)V, {z, g} ∈ L(λ), Imλ 6= 0 (44)
is equivalent to (43). By (44) and the equality L(λ̄)=L∗(λ), it follows that L(λ)
is continuously invertible. The pair {z, g} ∈ L(λ) if and only if the pair {z, h} ∈
LF (λ), where g =h−Zλz. Hence the relation LF (λ) is maximal accumulative for
Imλ>0 if and only if (44) holds and (LF (λ))∗ = LF (λ̄).
184 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2
On the Characteristic Operator of an Integral Equation...
Thus, if the function λ→F (λ) belongs to F(Ni, N−i), then the family λ→
L(λ) = LF (λ) − Zλ has the following properties for Imλ 6= 0: this family is
holomorphic; L(λ̄) = L∗(λ); the inequality (44) holds. From Theorems 5, 6, it
follows that the family λ→L(λ) generates a characteristic operator. The converse
statement is valid since the above argument is reversible. The theorem is proved.
Corollary 9. Let L(λ) be linear relations with the property L0(λ)⊂ L(λ)⊂
L∗0(λ̄), Imλ 6=0. The family λ→L(λ) generates a characteristic operator of equa-
tion (38) if and only if this family satisfies the following conditions: 1) L(λ̄) =
L∗(λ); 2) the family λ→L(λ) is holomorphic for Imλ 6=0; 3) the inequality
(Imλ)−1Im(g, z)V 6 −(Imλ)−1Im(Zλz, z)V = −
b0∫
a0
((dṼλ)z(t), z(t))dt
holds for Imλ 6= 0 and for all pairs {z, g} ∈ L(λ).
We notice that the statements, similar to Theorem 7 and Corollary 9, for a
differential equation with the Nevanlinna operator function are obtained in [4, 5]
in another way.
For the regular case, in terms of boundary values we will describe the families
generating the characteristic operator. According to Lemma 6, the pair {ỹ, f̃} ∈
H×H belongs to the relation L−λE if and only if there exists a pair {y, f} such
that the pairs {ỹ, f̃}, {y, f} are identical in H× H, and there holds the equality
y(t) = Wλ(t)ĉλ −Wλ(t)iJ
t∫
a0
W ∗̄
λ(s)(dṼ)f(s)ds,
where ĉλ ∈ Q−.
We take here λ = 0. To each pair {y, f} ∈ L, we assign a pair of boundary
values {Y, Y′} ∈ Q−×Q+ by the formulas
Y′ = w∗(0)f =
b0∫
a0
W∗
0 (s)(dṼ)f(s) ∈ Q+, Y = ĉ0 − (1/2)iJY′ ∈ Q−.
Let Γ, Γ1, Γ2 be the operators defined by the equalities Γ{y, f} = {Y,Y′},
Γ1{y, f} = Y, Γ2{y, f} = Y′. We apply Lemma 8 to the relation L. Then, using
(33), we get
(f, z)V − (y, g)V = (Y′, Z)− (Y,Z′), (45)
where {y, f}, {z, g} ∈ L, Γ{y, f} = {Y,Y′}, Γ{z, g} = {Z,Z′}. It follows from
Lemma 5 and Corollary 4 that the mapping Γ:L → Q−×Q+ is surjective. Thus
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2 185
V.M. Bruk
the quadruple (Q−, Q+, Γ1, Γ2) is the space of boundary values for the relation L
in the sense of [21, 22]. (In [21, 22], the case, where the relation is an operator
and Q− = Q+, is considered.)
R e m a r k 5. For the pairs {y,f}∈L′ by straightforward calculations, we
get
Y=2−1(W−1
0 (b0)y(b0)+W−1
0 (a0)y(a0)), Y′= iJ(W−1
0 (b0)y(b0)−W−1
0 (a0)y(a0)).
It follows from (45) and equality R(Γ) = Q−×Q+ that the pair {y, f} belongs
to L0 if and only if Γ{y, f} = 0. Consequently, between the relations L̂ with the
property L0 ⊂ L̂ ⊂ L and the relations ϑ ⊂ Q− × Q+ there exists a one-to-one
correspondence determined by the equality ΓL̂ = ϑ.
Denote by T (Q−, Q+) a set of holomorphic families of the linear relations
λ→ ϑ(λ) (Imλ 6= 0) whose values are the relations ϑ(λ) ⊂ Q− × Q+ such that
ϑ(λ̄) = ϑ∗(λ) and ϑ(λ) is maximal accumulative for each λ with Imλ > 0.
Theorem 8. Between the functions λ→F (λ) belonging to F(Ni, N−i) and the
families λ→ϑ(λ) belonging to T (Q−,Q+)there exists a one-to-one correspondence
determined by the equality ΓLF (λ) = ϑ(λ), i.e., the pair {y, f} ∈ L belongs to the
relation LF (λ) if and only if the pair Γ{y, f} = {Y,Y′} belongs to ϑ(λ).
P r o o f. It follows from the definitions of positive and negative spaces
(see [9, ch. 1], [11, ch. 2]) that there exist the operators I1, I2 such that I1,
I2 are isometrics Q+ onto Q and Q onto Q−, respectively, and the equality
(Y, Y′) = (I−1
2 Y, I1Y′) holds for all elements Y ∈ Q−, Y′ ∈ Q+. We define the
operator Γ̂ : L → Q × Q by the equality Γ̂{y, f} = {Ŷ, Ŷ′}, where {y, f} ∈ L,
Ŷ= I−1
2 Y, Ŷ′=I1Y′, {Y, Y′}=Γ{y, f}. Then the operator Γ̂ maps L onto Q×Q
and equality (45) holds when Y, Y′, Z, Z′ is replaced by Ŷ, Ŷ′, Ẑ, Ẑ′, respectively.
According to [22], a pair {y, f} ∈ L belongs to LF (λ) if and only if
(K(λ)− E)Ŷ′ − i(K(λ) + E)Ŷ = 0,
where λ → K(λ) is a holomorphic function for Imλ > 0 whose values are the
operators in Q with the norm ‖K(λ)‖6 1. The functions F and K determine
uniquely each other.
By ϑ̂(λ), denote the Cayley transformation of the operator K(λ), i.e., ϑ̂(λ) =
i(K(λ)+E)(K(λ)−E)−1. We set ϑ̂(λ̄)= ϑ̂∗(λ). The function λ→K(λ) has the
properties listed above if and only if the family of relations λ → ϑ̂(λ) belongs
T (Q,Q). On the other hand, the family of relations λ→ ϑ̂(λ) belongs T (Q,Q) if
and only if the family λ→ϑ(λ)= I−1
1 ϑ̂(λ)I−1
2 belongs T (Q−, Q+). The theorem
is proved.
186 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 2
On the Characteristic Operator of an Integral Equation...
It follows from (24) that {y, g} ∈ L(λ) if and only if {y, g + Zλy} ∈ L. We
set Γ(λ){y, g} = Γ{y, g + Zλy} for each pair {y, g} ∈ L(λ).
Theorem 8 implies the following statement.
Corollary 10. The family of the relations λ→L(λ), Imλ 6= 0, generates some
characteristic operator if and only if the family λ→Γ(λ)L(λ) = ϑ(λ) belongs to
T (Q−, Q+).
Notice that for the first time the linear relations were used to describe self-
adjoint extensions of differential operators in [23] (further bibliography can be
found in [11, 24]).
The author expresses his gratitude to the referee for useful remarks.
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