On Linear Relations Generated by a DiRerential Expression and by a Nevanlinna Operator Function
The families of maximal and minimal relations generated by a differential expression with bounded operator coe±cients and by a Nevanlinna operator function are defined. These families are proved to be holomorphic. In the case of finite interval, the space of boundary values is constructed. In terms...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2011
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| Цитувати: | On Linear Relations Generated by a DiRerential Expression and by a Nevanlinna Operator Function / V.M. Bruk // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 2. — С. 115-140. — Бібліогр.: 25 назв. — англ. |
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irk-123456789-1066682016-10-02T03:03:04Z On Linear Relations Generated by a DiRerential Expression and by a Nevanlinna Operator Function Bruk, V.M. The families of maximal and minimal relations generated by a differential expression with bounded operator coe±cients and by a Nevanlinna operator function are defined. These families are proved to be holomorphic. In the case of finite interval, the space of boundary values is constructed. In terms of boundary conditions, a criterion for the restrictions of maximal relations to be continuously invertible and a criterion for the families of these restrictions to be holomorphic are given. The operators inverse to these restrictions are stated to be integral operators. By using the results obtained, the existence of the characteristic operator on the finite interval and the axis is proved. 2011 Article On Linear Relations Generated by a DiRerential Expression and by a Nevanlinna Operator Function / V.M. Bruk // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 2. — С. 115-140. — Бібліогр.: 25 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106668 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The families of maximal and minimal relations generated by a differential expression with bounded operator coe±cients and by a Nevanlinna operator function are defined. These families are proved to be holomorphic. In the case of finite interval, the space of boundary values is constructed. In terms of boundary conditions, a criterion for the restrictions of maximal relations to be continuously invertible and a criterion for the families of these restrictions to be holomorphic are given. The operators inverse to these restrictions are stated to be integral operators. By using the results obtained, the existence of the characteristic operator on the finite interval and the axis is proved. |
| format |
Article |
| author |
Bruk, V.M. |
| spellingShingle |
Bruk, V.M. On Linear Relations Generated by a DiRerential Expression and by a Nevanlinna Operator Function Журнал математической физики, анализа, геометрии |
| author_facet |
Bruk, V.M. |
| author_sort |
Bruk, V.M. |
| title |
On Linear Relations Generated by a DiRerential Expression and by a Nevanlinna Operator Function |
| title_short |
On Linear Relations Generated by a DiRerential Expression and by a Nevanlinna Operator Function |
| title_full |
On Linear Relations Generated by a DiRerential Expression and by a Nevanlinna Operator Function |
| title_fullStr |
On Linear Relations Generated by a DiRerential Expression and by a Nevanlinna Operator Function |
| title_full_unstemmed |
On Linear Relations Generated by a DiRerential Expression and by a Nevanlinna Operator Function |
| title_sort |
on linear relations generated by a direrential expression and by a nevanlinna operator function |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106668 |
| citation_txt |
On Linear Relations Generated by a DiRerential Expression and by a Nevanlinna Operator Function / V.M. Bruk // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 2. — С. 115-140. — Бібліогр.: 25 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT brukvm onlinearrelationsgeneratedbyadirerentialexpressionandbyanevanlinnaoperatorfunction |
| first_indexed |
2025-07-07T18:50:22Z |
| last_indexed |
2025-07-07T18:50:22Z |
| _version_ |
1837015208528183296 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2011, vol. 7, No. 2, pp. 115–140
On Linear Relations Generated by a Differential
Expression and by a Nevanlinna Operator Function
V.M. Bruk
Saratov State Technical University
77, Politechnicheskaya Str., Saratov 410054, Russia
E-mail: vladislavbruk@mail.ru
Received April 22, 2010
The families of maximal and minimal relations generated by a differ-
ential expression with bounded operator coefficients and by a Nevanlinna
operator function are defined. These families are proved to be holomorphic.
In the case of finite interval, the space of boundary values is constructed.
In terms of boundary conditions, a criterion for the restrictions of maxi-
mal relations to be continuously invertible and a criterion for the families
of these restrictions to be holomorphic are given. The operators inverse to
these restrictions are stated to be integral operators. By using the results
obtained, the existence of the characteristic operator on the finite interval
and the axis is proved.
Key words: Hilbert space, linear relation, differential expression, holo-
morphic family of relations, resolvent, characteristic operator, Nevanlinna
function.
Mathematics Subject Classification 2000: 47A06, 47A10, 34B27.
Introduction
On a finite or infinite interval (a, b), we consider a differential expression
lλ[y] = l[y] − Cλy, where l is a formally self-adjoint differential expression with
bounded operator coefficients in a Hilbert space H, Cλ = Cλ(t) is a Nevanlinna
operator function, i.e., Cλ(t) is a holomorphic function for Imλ 6= 0 whose values
are the bounded operators in H such that C ∗̄
λ
(t) = Cλ(t) and Im Cλ(t)/Imλ > 0.
Let A(t)=Im Cλ0(t)/Imλ0. In the space L2(H, A(t); a, b), we define the fami-
lies L0(λ), L(λ) of minimal and maximal relations, respectively, and prove that
these families are holomorphic for Imλ 6= 0. In the case of finite interval, the space
of boundary values for the family of maximal relations L(λ) is constructed. There
The work is supported by the Russian Foundation of Basic Researches (grant 10-01-00276)
c© V.M. Bruk, 2011
V.M. Bruk
exists a one-to-one correspondence between the relations L̂(λ) with the property
L0(λ) ⊂ L̂(λ) ⊂ L(λ) and the boundary relations θ(λ). This correspondence is
determined by the equality γ(λ)L̂(λ) = θ(λ), where γ(λ) is the operator which
to each pair from L(λ) associates the pair of boundary values (in this case, we
denote L̂(λ)=Lθ(λ)(λ) ). We prove that 1) for fixed λ, the relation (Lθ(λ)(λ))−1
is a bounded everywhere defined operator if and only if the relation θ−1(λ) is
the operator with the same property; in this case the operator (Lθ(λ)(λ))−1 is
integral; 2) the operator function (Lθ(λ)(λ))−1 is holomorphic if and only if θ−1(λ)
is holomorphic. If the function Cλ is linear, these results correspond with the
description of the generalized resolvents from [1–3].
We apply the obtained results to prove the existence of the characte-
ristic operator on the finite interval and the axis. In the papers by V.I. Khra-
bustovsky [4–6], the definition of the characteristic operator is given. In these
papers, for a differential operator of first order the existence of the characteristic
operator is established and various problems associated with the characteristic
operator are studied. Bibliography on the characteristic operator is given in [4–6].
1. Main Assumptions and Auxiliary Statements
Let H be a separable Hilbert space with the scalar product (·, ·) and the norm
‖·‖. On a finite or infinite interval (a, b) we consider a differential expression l of
order r
l[y]=
n∑
k=1
(−1)k{(pn−k(t)y(k))(k)− i[(qn−k(t)y(k))k−1+ (qn−k(t)y(k−1))(k)]}+
+pn(t)y,
n∑
k=0
(−1)k{i[(qn−k(t)y(k))(k+1)+ (qn−k(t)y(k+1))k] + (pn−k(t)y(k))(k)},
where r = 2n (n = 1, 2, . . .) or r = 2n + 1 (n = 0, 1, 2, . . .). The coefficients of
l are the bounded self-adjoint operators in H. The leading coefficients, p0(t) in
the case r = 2n and q0(t) in the case r = 2n + 1, have bounded inverse operators
almost everywhere. According to [7], l is considered to be a quasidifferential
expression. The quasiderivatives for the expression l are defined in [7]. Suppose
that the functions pj(t), qk(t) are strongly continuous, the function q0(t) in the
case r = 2n + 1 is strongly differentiable, and the norms of the functions
p−1
0 (t), p−1
0 (t)q0(t), q0(t)p−1
0 (t)q0(t), p1(t), . . . , pn(t), q0(t), . . . , qn−1(t),
in case r = 2n,
q′0(t), q1(t), . . . , qn(t), p0(t), . . . , pn(t), in case r = 2n + 1,
are integrable on every compact interval [α, β] ⊂ (a, b).
116 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2
On Linear Relations Generated by a Differential Expression...
The boundary a is called a regular boundary if a > −∞, and one can take
α = a, and the norm ‖Cλ(t)‖ is integrable on every interval [a, β) (β < b), where
the function Cλ(t) is defined bellow. In the contrary case, the boundary a is called
a singular boundary. The regularity and singularity of the boundary b are defined
analogously. For brevity, we assume that if the interval (a, b) is finite, then the
boundaries a, b are regular. The singular case is studied on the example of the
axis (−∞,∞). Other cases of singular boundaries are considered analogously.
Let Cλ(t) be an operator function whose values are the bounded operators
in H. Suppose Cλ(t) satisfies the following conditions [4]:
(a) there exist sets C0⊃C \ R1 and I0⊂ (a, b) with the following properties:
the measure of the set (a, b)\ I0 equals zero and every point belonging to C0 has
a neighborhood independent of t∈I0 such that the function Cλ(t) is holomorphic
with respect to λ in this neighborhood;
(b) for all λ ∈ C0, the function Cλ(t) is Bochner locally integrable in the
uniform operator topology;
(c) C∗λ(t) = Cλ̄(t) as well as Im Cλ(t)/Imλ is a nonnegative operator for all
t ∈ I0 and for all λ such that Imλ 6= 0.
We denote Bλ(t) = Re Cλ(t), Aλ(t) = Im Cλ(t). Using condition (c), we get
Bλ̄ = Bλ, Aλ̄(t) = −Aλ(t). Let the operator aλ(t) be given by aλ(t) = Aλ(t)/Imλ
for λ such that Imλ 6= 0. It follows from condition (c) that for all µ ∈ C0 ∩ R
there exists limλ→µ±i0 aλ(t) = aµ(t). The function aµ(t) is locally integrable [4].
Further, we will use the following statements (see [4]).
Statement 1. If for some element g ∈ H there exist numbers t0 ∈ I0 and
λ0 ∈ C0 such that aλ0(t0)g = 0, then for all λ ∈ C0 the equalities aλ(t0)g = 0 and
(Bλ(t0)− Bλ0(t0))g = 0 hold.
Statement 2. Suppose t ∈ I0, λ, µ ∈ C0 and a sequence of the elements
gn∈H satisfy the condition (aλ(t)gn, gn)→0 as n→∞. Then (λ− µ)−1(Cλ(t)−
Cµ(t))gn→0.
Statement 3. For any compact K ⊂ C0 there exist constants k1, k2 > 0
such that for all h ∈ H, t ∈ I0, λ1, λ2 ∈ K the inequality k1(aλ1(t)h, h)1/2 6
(aλ2(t)h, h)1/2 6 k2(aλ1(t)h, h)1/2 holds (thus the constants k1, k2 are indepen-
dent of t∈I0, λ∈K).
We denote lλ[y] = l[y] − (Bλ(t) + iAλ(t))y = l[y] − Cλ(t)y. Let Wj,λ(t) be
the operator solution of the equation lλ[y] = 0 satisfying the initial conditions
W
[k−1]
j,λ (t0) = δj,kE, where t0 ∈ (a, b), δj,k is the Kronecker symbol, E is the iden-
tity operator, j, k = 1, . . . , r. By Wλ(t) we denote the operator one-row matrix
Wλ(t) = (W1,λ(t), . . . ,Wr,λ(t)). The operator Wλ(t) is a continuous mapping of
Hr into H. The adjoint operator W ∗
λ (t) maps continuously H into Hr. For fixed
t, the function Wλ(t) is holomorphic on C0.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2 117
V.M. Bruk
If lλ is defined for the function y, then we denote ŷ = (y[0], y[1], . . . , y[r−1])T
(T in the upper index denotes transposition). Let u = (u1, . . . , un) be a system of
functions such that lλ[uj ] exists for j = 1, . . . , n. By û we denote the matrix with
the jth column ûj (j = 1, . . . , n). Similar notations are used for the operator
functions. Note that all quasiderivatives up to order r − 1 inclusive coincide for
the expressions l and lλ.
We consider the operator matrices of orders 2n and 2n + 1 for the expression
l in the cases r = 2n and r = 2n + 1, respectively:
J2n(t) =
−E
...
−E
E
...
E
,
J2n+1(t) =
−E
...
−E
2iq−1
0 (t)
E
...
E
,
where all non indicated elements are equal to zero. (In matrix J2n+1 the element
2iq−1
0 (t) stands on the intersection of the row n + 1 and the column n + 1.)
Suppose the expression l is defined for the functions y, z, and l[y], l[z] are
locally integrable on (a, b). Then Lagrange’s formula has the form
β∫
α
(l[y], z)dt−
β∫
α
(y, l[z])dt = (Jr(t)ŷ(t), ẑ(t))|βα , a < α 6 β < b. (1)
In (1), we take y(t) = Wλ(t)c, z(t) = Wλ̄(t)d (c, d ∈ Hr). Since l[y] = (Bλ+iAλ)y,
l[z] = (Bλ − iAλ)z, we obtain
Ŵ ∗̄
λ (t)Jr(t)Ŵλ(t) = Jr(t0). (2)
It follows from (2) and from the ”method of the variation of arbitrary con-
stants” that the general solution of the equation lλ[y] = f(t) is represented in the
form
yλ(t, f) = Wλ(t)
c + J−1
r (t0)
t∫
t0
W ∗̄
λ (s)f(s)ds
, (3)
118 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2
On Linear Relations Generated by a Differential Expression...
where c ∈ Hr, the function f is locally integrable on (a, b). Consequently,
ŷλ(t, f) = Ŵλ(t)
c + J−1
r (t0)
t∫
t0
W ∗̄
λ (s)f(s)ds
. (4)
We fix some λ0 ∈ C0. For brevity, we denote A(t) = aλ0(t). On the set
of functions continuous and finite on the interval (a, b) and ranging in H, we
introduce the quasi-scalar product
(y, z)A =
b∫
a
(A(t)y(t), z(t))dt. (5)
We identify the functions such that (y, y)A = 0 with zero and perform the com-
pletion. Then we obtain a Hilbert space denoted by H = L2(H,A(t); a, b). The
elements of H are the classes of functions identified with each order in the norm
‖y‖A =
b∫
a
∥∥∥A1/2(t)y(t)
∥∥∥
2
dt
1/2
. (6)
In order not to complicate terminology, we denote the class of functions with
representative y by the same symbol. We will also say that the function y belongs
to H. We treat the equalities between functions belonging to H as the equalities
between corresponding equivalence classes.
It follows from Statements 1, 3 that the space H does not depend on the choice
of point λ ∈ C0 in the sense below. If we change A(t) = aλ0(t) either to aλ(t)
(λ ∈ C0) or to Aλ(t) (Imλ > 0) in (5), we obtain the same set H supplied with
the equivalent norm.
According to Statement 1, the set G(t) = ker aλ(t) (t ∈ I0) is independent
of λ ∈ C0. Let H(t) be the orthogonal complement of G(t) in H, i.e, H(t) =
HªG(t); and A0(t) be the restriction of A(t) to H(t). By H−1/2(t) we denote the
completion of H(t) with respect to the norm (A0(t)x, x)1/2. The space H−1/2(t)
can be treated as a space with negative norm with regard to H(t) [8, Ch. 2].
Let H1/2(t) denote the corresponding space with a positive norm. It follows
from Statements 1, 3 that the spaces H1/2(t), H−1/2(t) do not depend on the
replacement of A(t) = aλ0(t) by aλ(t) (λ ∈ C0) or by Aλ(t) (Imλ > 0) in the
sense below. After replacing we obtain the same sets H1/2(t), H−1/2(t), but the
norms on H1/2(t), H−1/2(t) are changed to the equivalent norms. Moreover, the
constants in the inequalities between the norms do not depend on t ∈ I0 and
λ ∈ K, where the compact K ⊂ C0.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2 119
V.M. Bruk
Suppose Imλ > 0. Let A1/2
0,λ (t), A0,λ(t) be the restrictions respectively of
A1/2
λ (t), Aλ(t) to H(t). It follows from the above argument and the definition of
positive and negative spaces that the operators A1/2
0,λ (t), A0,λ(t) possess the con-
tinuous extensions Ã1/2
0,λ (t), Ã0,λ(t) to H−1/2(t). The operator Ã1/2
0,λ (t) (Ã0,λ(t)) is
the continuous and one-to-one mapping of H−1/2(t) onto H(t) (onto H1/2(t), res-
pectively). Moreover, the equality Ã0,λ(t) = A1/2
0,λ (t)Ã1/2
0,λ (t) holds. The operator
A1/2
0,λ (t) is a bijection of H(t) onto H1/2(t). For Imλ<0, we set Ã0,λ(t)=−Ã0,λ̄(t).
Let Ãλ(t) (Imλ 6=0) denote the operator defined on H−1/2(t)⊕G(t) which is
equal to Ã0,λ(t) on H−1/2(t) and to zero on G(t). Obviously, Ãλ(t) is an extension
of Aλ(t). We apply the similar notations for the operators aλ(t) and A(t). In
particular, Ã(t) is an extension of A(t) to H−1/2(t)⊕G(t) obtained in the same
way as the extension Ãλ(t). The above argument and Statement 3 imply
∥∥∥Ãλ(t)Ã−1
0,µ(t)x
∥∥∥
H1/2(t)
6 k ‖x‖H1/2(t) , x ∈ H1/2(t),
∥∥∥Ãλ(t)x1
∥∥∥
H1/2(t)
6 k1 ‖x1‖H−1/2(t) , x1 ∈ H−1/2(t), (7)
where the constants k, k1 are independent of t∈I0. (Here and further, the sym-
bols k, k1, ... denote positive constants that are different in various inequalities,
in general.)
In [1], it was shown that the spaces H−1/2(t) are measurable with respect
to the parameter t [9, Ch. 1] whenever for the family of measurable functions
one takes the family of functions of the form Ã−1
0 (t)A1/2(t)h(t), where h(t) is a
measurable function with the values in H. The space H is a measurable sum
of the spaces H−1/2(t), and H consists of the elements (i.e., classes of functions)
with representatives of the form Ã−1
0 (t)A1/2(t)h(t), where h(t) ∈ L2(H; a, b) [1].
We denote Bλ,µ(t) = Bλ(t) − Bµ(t), where λ, µ ∈ C0, t ∈ I0. It follows
from Statement 1 that G(t) = ker aλ(t) ⊂ kerBλ,µ(t). Therefore, R(Bλ,µ(t)) ⊂
R(aλ(t)) = H(t). By B(0)
λ,µ(t) denote the restriction of Bλ,µ(t) to H(t).
Lemma 1. For all t ∈ I0, λ, µ ∈ C0, the operator B(0)
λ,µ(t) possesses the
continuous extension B̃(0)
λ,µ(t) onto the space H−1/2(t). The operator B̃(0)
λ,µ(t) maps
H−1/2(t) into H1/2(t) continuously.
P r o o f. Following [4], we represent the function Cλ(t) in the form
Cλ(t) = Re Ci(t) + λU(t) +
∞∫
−∞
(
1
τ − λ
− τ
1 + τ2
)
dσt(τ), (8)
120 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2
On Linear Relations Generated by a Differential Expression...
where U(t) are nonnegative bounded operators in H, σt(τ) is a nondecreasing
operator function for each fixed t ∈ I0 whose values are bounded operators in H
and
∫∞
−∞
d(σt(τ)g,g)
1+τ2 < ∞ for any g ∈ H. Then
aλ(t) = U(t) +
+∞∫
−∞
dσt(τ)
|τ − λ|2 . (9)
Using (8), (9), we obtain
∣∣(λ− µ)−1((Cλ(t)− Cµ(t))g, h)
∣∣ 6
∥∥∥U1/2(t)g
∥∥∥
∥∥∥U1/2(t)h
∥∥∥
+
∞∫
−∞
(dσt(τ)g, g)
|t− λ|2
1/2
∞∫
−∞
(dσt(τ)h, h)
|t− µ|2
1/2
6
∥∥∥U1/2(t)g
∥∥∥
2
+
∞∫
−∞
(dσt(τ)g, g)
|t− λ|2
1/2
∥∥∥U1/2(t)h
∥∥∥
2
+
∞∫
−∞
(dσt(τ)h, h)
|t− µ|2
1/2
= (aλ(t)g, g)1/2(aµ(t)h, h)1/2 =
∥∥∥a
1/2
λ (t)g
∥∥∥
∥∥∥a1/2
µ (t)h
∥∥∥
for all g, h ∈ H. Hence, |(Bλ,µ(t)g, h)| 6 |λ− µ|
∥∥∥a
1/2
λ (t)g
∥∥∥
∥∥∥a
1/2
µ (t)h
∥∥∥. According
to Statement 3, there exists a constant k > 0 such that k is independent of t ∈ I0,
and
|(Bλ,µ(t)g, h)| 6 k
∥∥∥A1/2(t)g
∥∥∥
∥∥∥A1/2(t)h
∥∥∥ (k > 0). (10)
It follows from (10) that the operator B(0)
λ,µ(t) : H(t) → H(t) possesses the
continuous extension B̃(0)
λ,µ(t) : H−1/2(t)→H(t) ( this fact also follows from Sta-
tement 2). Let B̃+
λ,µ(t) be the adjoint operator for B̃(0)
λ,µ(t). Then B̃+
λ,µ(t) maps
H(t) into H1/2(t) continuously. We claim that B̃+
λ,µ(t) = B(0)
λ,µ(t). Since the
operator Bλ,µ(t) is self-adjoint, we have (Bλ,µ(t)x1, x2) = (x1,Bλ,µ(t)x2) for all
x1, x2 ∈ H(t). On the other hand,
(Bλ,µ(t)x1, x2) = (B̃0
λ,µ(t)x1, x2) = (x1, B̃+
λ,µ(t)x2).
Hence, B(0)
λ,µ(t) = B̃+
λ,µ(t). Thus, B(0)
λ,µ(t) maps H(t) into H1/2(t) continuously.
From this, we obtain that inequality (10) holds for all h ∈ H−1/2(t). Conse-
quently,
∥∥∥B(0)
λ,µ(t)g
∥∥∥
2
H1/2(t)
= (Ã−1/2
0 (t)Bλ,µ(t)g, Ã
−1/2
0 (t)Bλ,µ(t)g)
6 k
∥∥∥Ã
1/2
0 (t)g
∥∥∥
∥∥∥Ã
−1/2
0 (t)B(0)
λ,µ(t)g
∥∥∥ = k ‖g‖H−1/2(t)
∥∥∥B(0)
λ,µ(t)g
∥∥∥
H1/2(t)
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2 121
V.M. Bruk
for all g ∈ H(t). Hence,
∥∥∥B(0)
λ,µ(t)g
∥∥∥
H1/2(t)
6 k ‖g‖H−1/2(t) , (11)
where k > 0 is independent of t ∈ I0. All assertions of the lemma follow from
(11). Lemma 1 is proved.
Let B̃λ,µ(t) denote the operator defined on H−1/2(t)⊕G(t) which is equal to
B̃(0)
λ,µ(t) on H−1/2(t) and to zero on G(t). We set C̃λ,µ(t)= B̃λ,µ(t)+iÃλ(t)−iõ(t).
It follows from (7), (11) that C̃λ,µ(t), B̃λ,µ(t), Ãλ(t) are continuous mappings of
H−1/2(t)⊕G(t) into H1/2(t). By C̃λ,µ, B̃λ,µ, Ãλ denote the operators
y(t)→Ã−1
0 (t)C̃λ,µ(t)y(t), y(t)→Ã−1
0 (t)B̃λ,µ(t)y(t), y(t)→Ã−1
0 (t)Ãλ(t)y(t),
respectively. The operators C̃λ,µ, B̃λ,µ, Ãλ are considered in the space H. Thus,
with equalities (7), (11) being taken into account, we obtain the following corol-
laries from Lemma 1. (Corollary 1 is the generalization of Statement 2.)
Corollary 1. For all fixed λ, µ ∈ C0, t ∈ I0, the operator C̃λ,µ(t) maps
H−1/2(t) into H1/2(t) continuously and the inequality
∥∥∥C̃λ,µ(t)x
∥∥∥
H1/2(t)
6k ‖x‖H−1/2(t)
holds, where x ∈ H−1/2(t), the constant k is independent of t ∈ I0.
Corollary 2. For all fixed λ, µ ∈ C0, the operator C̃λ,µ is bounded in H. For
fixed λ or for fixed µ, the operator function C̃λ,µ is holomorphic with respect to
the other variable.
Corollary 3. For all fixed λ, µ ∈ C0, the operators Ãλ, B̃λ,µ are bounded
in H. The operator functions Ãλ, B̃λ,µ are continuous with respect to λ, µ ∈ C0
in the uniform operator topology.
2. Families of Maximal and Minimal Relations
Let B1, B2 be Banach spaces. The linear relation T is understood as any
linear manifold T ⊂ B1 ×B2. Terminology on the linear relations can be found,
for example, in [8], [10], [11]. From now onwards, the following notation are
used: {·, ·} is an ordered pair; 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 the
domain of T; R(T) is the range of T. The relation T is called invertible if T−1 is
an operator and T is called continuously invertible if T−1 is a bounded everywhere
defined operator. The linear operators are regarded as linear relations, and so the
notations S ⊂ B1 ×B2, {x1, x1} ∈ S are used for the operators S. Let B1 = B2
122 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2
On Linear Relations Generated by a Differential Expression...
be a Hilbert space. A linear relation T is called accumulative (dissipative) if for
all pairs {x1, x2} ∈ T the inequality Im(x2, x1) 6 0 (Im(x2, x1) > 0) holds. The
accumulative (dissipative) relation is called maximum if it does not have proper
accumulative (dissipative) extensions. Since all the relations considered further
are linear, the word ”linear” will often be omitted.
A family of the linear manifolds in a Banach space B is understood as a
function λ → T (λ), where T (λ) is a linear manifold, T (λ) ⊂ B. A family of
(closed) subspaces T (λ) is called holomorphic at the point λ1 ∈ C if there exists
a Banach space B0 and a family of the bounded linear operators K(λ) :B0→B
such that the operator K(λ) bijectively maps B0 onto T (λ) for any fixed λ and
the family λ → K(λ) is holomorphic in some neighborhood of λ1. A family of
subspaces is called holomorphic on the domain D if it is holomorphic at all points
belonging to D. Since the closed relation T(λ) is the subspace in B1 × B2, the
definition of holomorphic families is applied to the families of linear relations.
This definition generalizes the corresponding definition of holomorphic families
of closed operators [12, Ch. 7].
In [13], [14], the statement below is proved for the families of closed relations.
However, the proof remains valid for the families of subspaces.
Statement 4. Let T (λ)⊂B be a family of subspaces in a Banach space B.
Suppose there exists a subspace N⊂B such that the decomposition in the direct
sum B=T (λ1)uN holds for some fixed point λ1. The family T (λ) is holomorphic
at λ1 if and only if the space B is decomposed in the direct sum of subspaces T (λ)
and N for all λ belonging to some neighborhood of λ1 and the family P(λ) is holo-
morphic at λ1, where P(λ) is the projection of space B onto T (λ) in parallel N.
The proof of the following statement is given in [15].
Statement 5. Let T (λ) be a family of linear relations holomorphic at the
point λ1, and S(λ) be an operator function holomorphic at the point λ1 whose
values are the bounded everywhere defined operators. Then the family of relations
T (λ) + S(λ) is holomorphic at λ1.
Let Q0 be the set of elements c ∈ Hr such that the function Wµ(t)c (µ ∈ C0)
is identified with zero in H, i.e., the equality ‖A(t)Wµ(t)c‖ = 0 holds almost
everywhere. Using (3), we get
Wµ(t)c = Wλ(t)c
+ Wλ(t)J−1
r (t0)
t∫
t0
W ∗̄
λ (s)(Bµ(s)−Bλ(s)+iAµ(s)− iAλ(s))Wµ(s)cds. (12)
It follows from (12) and Statement 1 that the set Q0 does not change if we
substitute Wλ for Wµ (λ, µ ∈ C0).
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2 123
V.M. Bruk
Rema r k 1. If we change λ to µ and µ to λ in (12), then we obtain the true
equality (λ, µ ∈ C0).
Further, in this section and Section 3, we assume that the boundaries a, b are
regular.
Let Q be the orthogonal complement of Q0 in Hr, i.e., Q = Hr ªQ0. In Q,
we introduce the norm
‖c‖− =
b∫
a
∥∥∥A1/2(s)Wµ(s)c
∥∥∥
2
ds
1/2
6 k ‖c‖ , c ∈ Q, k > 0. (13)
We denote the completion of Q with respect to this norm by Q−. By (12), we get
∥∥∥A1/2(t)Wλ(t)c
∥∥∥
6
∥∥∥A1/2(t)Wµ(t)c
∥∥∥ + k
∥∥∥A1/2(t)
∥∥∥
b∫
a
‖(Cλ(s)− Cµ(s))Wµ(s)c‖ ds.
We will obtain the analogous estimate for the norm
∥∥A1/2(t)Wµ(t)c
∥∥ if we change
Wλ to Wµ and Wµ to Wλ. It follows from this and Statement 2 that the replace-
ment of Wµ by Wλ in (13) leads to the same set Q− with the equivalent norm.
Thereby the space Q− is independent of λ ∈ C0. The space Q− can be treated
as a space with negative norm with respect to Q [8, Ch. 2]. By Q+ denote the
corresponding space with positive norm.
The operator c → Wλ(t)c is a continuous one-to-one mapping of Q− into H.
We denote this operator by Wλ. Its range is closed in H. Consequently, the
adjoint operator W∗
λ continuously maps H onto Q+. We find the form W∗
λ. For
any elements x ∈ Q and f ∈ H, we have
(f,Wλx)A =
b∫
a
(Ã(s)f(s),Wλ(s)x)ds =
b∫
a
(W ∗
λ (s)Ã(s)f(s), x)ds = (W∗
λf, x).
Hence, taking into account that Q can be densely embedded in Q−, we obtain
W∗
λf =
b∫
a
W ∗
λ (s)Ã(s)f(s)ds.
Here we replace λ by λ̄ and summarize the properties of operators Wλ, W ∗̄
λ
in
the following statement.
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Lemma 2. For fixed λ ∈ C0, the operator Wλ is a continuous one-to-one
mapping of Q− into H, and the range R(Wλ) is closed. The operator W ∗̄
λ
maps
continuously H onto Q+. The operator functions Wλ, W ∗̄
λ
are holomorphic on C0.
In the same way as in [1], [3], we define the minimal and maximal relations
generated by the expression lλ (λ ∈ C0) and function A(t). Let D′
λ be a set
of functions y ∈ H satisfying the following conditions: (a) the quasiderivatives
y[k] exist, they are absolutely continuous up to the order r − 1 inclusively, and
lλ[y](t) ∈ H1/2(t) almost everywhere; (b) the function Ã−1
0 (t)lλ[y] ∈ H. To each
class of the functions identified in H with y ∈ D′
λ we assign the class of functions
identified in H with Ã−1
0 (t)lλ[y]. Thus, we get a linear relation L′(λ) ⊂ H × H.
Denote its closure by L(λ) and call it a maximal relation. We define the minimal
relation L0(λ) as a restriction of L(λ) to the set of functions y ∈ D′
λ satisfying
the conditions y[k](a) = y[k](b) = 0 (k = 0, 1, . . . , r − 1).
The following statements can be proved by analogy with the corresponding
assertions in [3], [16].
Lemma 3. For fixed λ ∈ C0, the relation L(λ) consists of all pairs {y, f} ∈
H× H for which
y = Wλ(t)cλ + Fλ, (14)
where cλ ∈ Q−, Fλ(t) = Wλ(t)J−1
r (t0)
t∫
a
W ∗̄
λ
(s)Ã(s)f(s)ds. The pair {y, f} ∈ L(λ)
belongs to L0(λ) if and only if cλ = 0 and
b∫
a
W ∗̄
λ (s)Ã(s)f(s)ds = 0. (15)
Corollary 4. The range R(L0(λ)) is closed and consists of the elements f ∈ H
with property (15). The relation L−1
0 (λ) is a bounded operator on its domain of
definition.
Corollary 5. The relation L0(λ) is closed.
Corollary 6. The operator Wλ is a continuous one-to-one mapping of Q−
onto kerL(λ).
Rema r k 2. In equality (14), the element cλ ∈ Q− and the function Fλ ∈ H
are uniquely determined by the pair {y, f} ∈ L(λ).
Theorem 1. For all λ, µ ∈ C0, the equality L(µ)=L(λ) + C̃λ,µ holds.
P r o o f follows from the definition of L(λ) and from Corollary 2.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2 125
V.M. Bruk
Corollary 7. The families of relations L(λ), L0(λ) are holomorphic on C0.
P r o o f follows from Theorem 1, Corollary 2, Statement 5.
Lemma 4. The equality (L0(λ))∗ = L(λ̄) holds for each λ ∈ C0.
Proo f. Let L̃(λ) be the maximal relation and L̃0(λ) be the minimal relation
generated by the formally self-adjoint expression l[y]−Bλ(t)y. In [1], [16], it was
shown that (L̃0(λ))∗ = L̃(λ) = L̃(λ̄). Using the equalities L(λ̄) = L̃(λ̄) − iÃλ̄,
L0(λ) = L̃0(λ)− iÃλ and Corollary 3, we obtain the desired assertion. Lemma 4
is proved.
R e m a r k 3. Taking into account Theorem 1 and the proof of Lemma 4, we
get D(L(µ)) = D(L(λ)) = D(L̃(λ)) and D(L0(µ)) = D(L0(λ)) = D(L̃0(λ)) for all
λ, µ ∈ C0.
Lemma 5. For fixed λ (Imλ 6= 0) and for all pairs {y0, g} ∈ L0(λ), the
inequality |Imλ|−1 Im(g, y0)A 6 −k ‖y0‖2
A , k > 0, holds.
P r o o f. Suppose Imλ>0. Using (1) and the definition of L0(λ), we get
Im(g, y0)A = −
b∫
a
(Aλ(t)y0(t), y0(t))dt. (16)
Now the desired assertion follows from (16) and Statement 3. The case Imλ < 0
is considered analogously. Lemma 5 is proved.
So, the relation L0(λ) is accumulative in the upper half-plane and L0(λ) is
dissipative in the lower half-plane.
3. Holomorphic Families of Invertible Restrictions of Maximal
Relations
Let B1, B2, B̃1, B̃2 be Banach spaces, T ⊂ B1 × B2 be a closed linear
relation, γ : T → B̃1 × B̃2 be a linear operator, γi = Piγ (i = 1, 2), where Pi is
the projection B̃1 × B̃2 onto B̃i, i.e., Pi{x1, x2} = xi. The following definition is
given in [17] for the operators and in [18] for the relations.
Definition 1. The quadruple (B̃1, B̃2, γ1, γ2) is called a space of boundary
values (SBV) for a closed relation T if the operator γ is a continuous mapping
of T onto B̃1× B̃2 and the restriction of the operator γ1 to KerT is a one-to-one
mapping of KerT onto B̃1.
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On Linear Relations Generated by a Differential Expression...
We define an operator Φγ : B̃1 → B̃2 by the equality Φγ = γ2β, where β =
(γ1|KerT )−1 is the operator inverse to the restriction of γ1 to KerT . We denote
T0 = ker γ. It follows from the definition of SBV that between the relations
θ ⊂ B̃1 × B̃2 and the relations T̃ with property T0 ⊂ T̃ ⊂ T there is a one-to-
one correspondence determined by the equality γT̃ = θ. In this case we denote
T̃ = Tθ. The relation θ is called a boundary relation. The following assertion is
proved in [18].
Statement 6. The relation Tθ is continuously invertible if and only if the
relation θ − Φγ is continuously invertible.
Suppose {y, f} ∈ L(λ) (λ ∈ C0). Then (14) holds. 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 ∗̄
λf =
b∫
a
W ∗̄
λ (s)Ã(s)f(s)ds, Yλ = cλ + (1/2)J−1
r (t0)Y ′λ. (17)
It follows from Remark 2 that the pair {Yλ,Y ′λ} of boundary values is uniquely
determined for each pair {y, f} ∈ L(λ). 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. Lemmas 2, 3 and Corollary 6 imply the following assertion.
Theorem 2. The quadruple {Q−, Q+, γ1(λ), γ2(λ)} is SBV for the relation
L(λ). The corresponding operator Φγ(λ) equals zero. Moreover, ker γ(λ) = L0(λ).
Theorem 3. The range R(γ(λ)) of the operator γ(λ) coincides with Q−×Q+,
and for all pairs {y, f}∈L(λ), {z, g}∈L(λ̄) (λ∈C0) ”the Green formula” is valid
(f, z)A − (y, g)A = (Y ′λ,Zλ̄)− (Yλ,Z ′̄λ), (18)
where {Yλ,Y ′λ} = γ(λ){y, f}, {Zλ̄,Z ′̄
λ
} = γ(λ̄){z, g}.
P r o o f. Using Lemmas 2, 3, we get R(γ(λ)) = Q−× Q+. According to
Lemma 3, y = Wλ(t)cλ + Fλ, z = Wλ̄(t)dλ̄ + Gλ̄, where cλ, dλ̄ ∈ Q−,
Fλ(t) = Wλ(t)J−1
r (t0)
t∫
a
W ∗̄
λ (s)Ã(s)f(s)ds,
Gλ̄(t) = Wλ̄(t)J−1
r (t0)
t∫
a
W ∗
λ (s)Ã(s)g(s)ds.
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V.M. Bruk
The operations lλ, lλ̄ can be applied to the functions Fλ, Gλ̄ and lλ[Fλ] =
Ã(t)f , lλ̄[Gλ̄] = Ã(t)g. From this, (1), (2), and (17), we obtain
(f, Gλ̄)A−(Fλ, g)A =(Jr(b)Ŵλ(b)J−1
r (t0)Y ′λ, Ŵλ̄(b)J−1
r (t0)Z ′̄λ)=(Y ′λ, J−1
r (t0)Z ′̄λ).
(19)
We take two sequences {cλ,n}, {dλ̄,n} such that cλ,n, dλ,n ∈ Q and {cλ,n}, {dλ̄,n}
converge to cλ ∈ Q− and dλ̄ ∈ Q− in Q−, respectively. We denote vn(t) =
Wλ(t)cλ,n, un(t) = Wλ̄(t)dλ̄,n. Then the sequences {vn(t)}, {un(t)} converge to
v(t) = Wλ(t)cλ, u(t) = Wλ̄(t)dλ̄ in H, respectively. Using (1), (2), (17), we obtain
(f, un)A =(f, un)A−(Fλ, 0)A =(Jr(t0)Ŵλ(b)J−1
r (t0)Y ′λ, Ŵλ̄(b)dλ̄,n)=(Y ′λ, dλ̄,n),
(vn, g)A =(vn, g)A−(0, Gλ̄)A =(Jr(t0)Ŵλ(b)cλ,n, Ŵλ̄(b)J−1
r (t0)Z ′̄λ)=−(cλ,n,Z ′̄λ).
In these equalities, we pass to the limit as n→∞. We get
(f, u)A = (Y ′λ, dλ̄), (v, g)A = −(cλ,Z ′̄λ). (20)
Using (19), (20), and the equality J∗r (t0) = −Jr(t0), we obtain
(f, z)A − (y, g)A = (Y ′λ, dλ̄)− (cλ,Z ′̄λ) + (Y ′λ, J−1
r (t0)Z ′̄λ) =
= (Y ′λ, dλ̄+(1/2)J−1
r (t0)Z ′̄λ)−(cλ+(1/2)J−1
r (t0)Y ′̄λ,Z ′̄λ) = (Y ′λ,Zλ̄)−(Yλ,Z ′̄λ).
Theorem 3 is proved.
Notice that for the first time the linear relations were applied to describe self-
adjoint extensions of differential operators in [19]. Further bibliography can, for
example, be found in [8], [10]. The boundary values in form (17) are suggested
in [16].
For fixed λ∈C0, between the relations L̂(λ) with the property L0(λ)⊂ L̂(λ) ⊂
L(λ) and relations θ ⊂ Q−×Q+ there is a one-to-one correspondence determined
by the equality γ(λ)L̂(λ) = θ. In this case we denote L̂(λ) = Lθ(λ). Thus,
a pair {y, f} ∈ Lθ(λ) if and only if {y, f} ∈ L(λ) and {Yλ,Y ′λ} ∈ θ, where
{Yλ,Y ′λ} = γ(λ){y, f}. Using Lemma 4, we get L∗(λ) = L0(λ̄), (L0(λ))∗ = L(λ̄).
Suppose L0(λ) ⊂ Lθ(λ) ⊂ L(λ). Then L0(λ̄) ⊂ (Lθ(λ))∗ ⊂ L(λ̄). Hence, taking
(18) into account, we obtain
(Lθ(λ))∗ = Lθ∗(λ̄), λ ∈ C0. (21)
Lemma 6. Let θ(λ) ⊂ Q−×Q+ be a family of linear relations defined on a set
D1 ⊂ C0 symmetric with respect to the real axis. Then the equality θ∗(λ)= θ(λ̄)
holds if and only if the equality (Lθ(λ)(λ))∗=Lθ(λ̄)(λ̄) holds.
P r o o f follows from equality (21).
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Theorem 4. For fixed λ ∈ C0, the relation Lθ(λ)(λ) is continuously invertible
if and only if so is the relation θ(λ). In this case, the operator Rλ = L−1
θ(λ)(λ) is
integral
Rλg =
b∫
a
Kλ(t, s)Ã(s)g(s)ds, g ∈ H,
where Kλ(t, s) = Wλ(t)(θ−1(λ)− (1/2)sgn(s− t)J−1
r (t0))W ∗̄
λ
(s).
Proof. The first part of the theorem follows from Statement 6 and Theorem 2.
Let us prove the second part of the theorem. For any pair {y, g} ∈ L(λ), we
transform equality (14) to the form
y(t) = Wλ(t)
c̃λ + (1/2)J−1
r (t0)
t∫
a
W ∗̄
λ (s)Ã(s)g(s)ds
−(1/2)J−1
r (t0)
b∫
t
W ∗̄
λ (s)Ã(s)g(s)ds
, (22)
where c̃λ = cλ + (1/2)J−1
r (t0)
∫ b
a W ∗̄
λ
(s)Ã(s)g(s)ds = Yλ = γ1(λ){y, g}.
Using (17), (22), we get
y(t) = Wλ(t)
θ−1(λ)Y ′λ + (1/2)J−1
r (t0)
t∫
a
W ∗̄
λ (s)Ã(s)g(s)ds
−(1/2)J−1
r (t0)
b∫
t
W ∗̄
λ (s)Ã(s)g(s)ds
.
This implies the desired assertion. Theorem 4 is proved.
Corollary 8. The equalities θ∗(λ)= θ(λ̄) and R∗
λ = Rλ̄ hold together.
Theorem 5. The function Rλ is holomorphic at the point λ1 ∈ C0 if and only
if the function θ−1(λ) is holomorphic at the same point.
P r o o f. According to Lemma 2 and Theorem 4, if the function θ−1(λ) is
holomorphic at λ1, then the function Rλ has the same property. Now suppose
that the function Rλ is holomorphic at λ1. Using Lemma 2 and Theorem 4, we
obtain that the function Wλθ−1(λ)W ∗̄
λ
g is holomorphic for every g ∈ H. It follows
from Remark 1 and equality (12) with the element c replaced by θ−1(λ)W ∗̄
λ
g that
the functions Wλθ−1(λ)W ∗̄
λ
g and Wµθ−1(λ)W ∗̄
λ
g are holomorphic together for
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2 129
V.M. Bruk
fixed µ ∈ C0. From Corollary 6 we conclude that the function θ−1(λ)W ∗̄
λ
g is
holomorphic at λ1 for any g ∈ H. Now the property that the function θ−1(λ) is
holomorphic at λ1 follows from Statement 7 proved in [20]. In this statement it
should be taken that B1 = H, B2 = Q+, B3 = Q−, S1(λ) = W ∗̄
λ
, S2(λ) = θ−1(λ),
S3(λ) = θ−1(λ)W ∗̄
λ
. Theorem 5 is proved.
Statement 7. Let B1, B2, B3 be Banach spaces. Suppose the bounded op-
erators S3(λ) : B1 →B3, S1(λ) : B1 →B2, S2(λ) : B2 →B3 satisfy the equality
S3(λ) = S2(λ)S1(λ) for every fixed λ belonging to some neighborhood of a point
λ1; moreover, R(S1(λ1)) = B2. If the functions S1(λ), S3(λ) are strongly dif-
ferentiable at the point λ1, then at the same point the function S2(λ) is strongly
differentiable.
Let Nµ =HªR(L0(µ)), where Imµ > 0. Using Lemmas 3, 4 and Corollary 6,
we get Nµ =kerL(µ̄). It follows from Theorem 1 that Nµ⊂D(L(λ)) and the pair
{w(t), Ã−1
0 (t)C̃µ̄,λ(t)w(t)} ∈ L(λ), (23)
where w(t) ∈ Nµ. The set D(L0(λ)) = D(L0(µ̄)) and Nµ are linearly independent
in H. For Imλ > 0, we define the relation L(λ) as the restriction of L(λ) to
D(L0(λ)) u Nµ. By (23) and the equality Aµ̄ = −Aµ, it follows that the relation
L(λ) consists of all pairs of the form
{y0,λ(t) + w(t), y1,λ(t) + Ã−1
0 (t)(B̃µ,λ(t)− i(Ãλ(t) + õ(t)))w(t)}, (24)
where {y0,λ, y1,λ} ∈ L0(λ), w(t) ∈ Nµ. For Imλ < 0, we set L(λ) = L∗(λ̄).
Theorem 6. The family of relations L(λ) is holomorphic for Imλ 6= 0.
P r o o f. First, we consider the case Imλ > 0. By Corollary 7, the family of
relations L0(λ) is holomorphic at every point λ1 (Imλ1 > 0). It follows from the
definition of holomorphic family that there exists a Banach space B and a family
of bounded linear operatorsK0(λ) :B→H such that the operatorK0(λ) bijectively
maps B onto L0(λ) for any fixed λ and the family λ → K0(λ) is holomorphic in
some neighborhood of λ1. By K1(λ) we denote an operator taking every element
w ∈ Nµ to the pair {w, C̃µ̄,λw} ∈ H × H. By Corollary 2, the operator K1(λ) is
bounded for fixed λ ∈ C0 and the family of operators K1(λ) is holomorphic on
C0. Consider the space B̃ = B ×Nµ and define the operator K̃(λ) : B̃ →H × H
by the formula K̃(λ){u,w} = K0(λ)u +K1(λ)w, where u ∈ B, w ∈ Nµ. For fixed
λ, the operator K̃(λ) is a continuous one-to-one mapping of B̃ onto L(λ) and the
family λ→K̃(λ) is holomorphic in some neighborhood of λ1. So, the theorem is
valid for Imλ > 0.
130 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2
On Linear Relations Generated by a Differential Expression...
Suppose Imλ < 0. Let the operator J : H×H→H × H be given by the for-
mula J {x, x1}= {−x1, x}. Then L∗(λ̄)=JL⊥(λ̄) (Here and further, we denote
N⊥=HªN , where N is a subspace of a Hilbert space H.) The family of subspa-
ces JL⊥(λ̄) is holomorphic if and only if the family L⊥(λ̄) is holomorphic. Now
the desired assertion is obtained from Lemma 7 below. Theorem 6 is proved.
Corollary 9. The family of relations L−1(λ) is holomorphic for Imλ 6= 0.
Lemma 7. Let H be a Hilbert space. If the family of subspaces T (λ) ⊂ H is
holomorphic at the point λ1, then the family T ⊥(λ̄) is holomorphic at λ̄1.
Proof. By Statement 4, H=T (λ)uN and the family P(λ) is holomorphic at
λ1, where P(λ) is the projection of spaceH onto T (λ) in parallel N. ThenQ(λ) =
E − P(λ) is the projection H onto N in parallel T (λ). Therefore, kerQ∗(λ) =
N⊥, R(Q∗(λ)) = T ⊥(λ), and (Q∗(λ))2 = Q∗(λ). Consequently, Q∗(λ) is the
projection of H onto T ⊥(λ) in parallel N⊥. The corresponding decomposition
has the form H = T ⊥(λ)uN⊥. Since the family Q∗(λ̄) is holomorphic, the family
of subspaces T ⊥(λ̄) is holomorphic at the point λ̄1. Lemma 7 is proved.
Theorem 7. For fixed λ, the relation L(λ) is maximal accumulative
if Imλ > 0 and it is maximal dissipative if Imλ < 0. The relation L−1(λ) is
a bounded everywhere defined operator.
Proo f. Let Imλ > 0, {z, z1} ∈ L(λ). Then {z, z1} has form (24). Using (1),
(24) and the equalities lµ̄[w] = 0, l[y0,λ] = Ã0y1,λ + iAλy0,λ + Bλy0,λ, we get
(z1, z)A = (y1,λ, y0,λ)A + (y1,λ, w)A + (B̃µ,λw, y0,λ)A + (B̃µ,λw,w)A
− i((Ãλ + õ)w, y0,λ)A − i((Ãλ + õ)w,w)A
= (y1,λ, y0,λ)A + (y1,λ, w)A + (w, y1,λ)A + (B̃µ,λw, w)A
− 2i(Ãλw, y0,λ)A − i((Ãλ + õ)w,w)A.
Hence, taking (16) into account, we obtain
Im(z1, z)A = Im(y1,λ, y0,λ)A − 2Re(Ãλw, y0,λ)A − ((Ãλ + õ)w,w)A
= −(Ãλ(y0,λ + w), y0,λ + w)A − (õw,w)A = −(Ãλz, z)A − (õw,w)A
= −
b∫
a
(Ãλ(t)z(t), z(t))dt−
b∫
a
(õ(t)w(t), w(t))dt.
Therefore,
Im(z1, z)A 6 −
b∫
a
(Ãλ(t)z(t), z(t))dt 6 −k(z, z)A (k > 0). (25)
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2 131
V.M. Bruk
Thus, the relation L(λ) is accumulative. It follows from (25) that the range
R(L(λ)) is closed and the relation L−1(λ) is the operator. Moreover,
R(L(µ)) = H. The equality R(L(λ)) = H is obtained from Lemma 8 and
Corollary 10 below. Consequently, the relation L(λ) is maximal accumulative
if Imλ > 0. For Imλ < 0, the desired assertion is obtained from the equality
L(λ)=L∗(λ̄). Theorem 7 is proved.
Lemma 8. Let H be a Hilbert space and S(λ) ⊂ H×H be a family of linear
relations holomorphic at the point µ. Suppose that the relation S(µ) has the
following properties: R(S(µ)) is a closed subspaces in H and the relation S−1(µ)
is an operator. Then there exists a neighborhood of µ such that the relations
S(λ) have the same properties for all points λ belonging to this neighborhood.
Moreover, dimHªR(S(λ)) = dimHªR(S(µ)).
P r o o f. We denote H0 =H ª R(S(µ)), N = H× H0. We claim that the
decomposition in the direct sum H × H = S(µ) u N holds. Indeed, any pair
{x1, x2} ∈ H ×H is uniquely produced by the form
{x1, x2} = {S−1(µ)Px2, Px2}+ {x1 − S−1(µ)Px2, P
⊥x2},
where P is the orthogonal projection H onto R(S(µ)), P⊥ = E − P .
According to Statement 4, the decomposition in the direct sum
H×H = S(λ) u N (26)
holds for all λ belonging to some neighborhood of µ. Using (26), we obtain
{x1, x2}={f1, f2}+{x1−f1, x2−f2} for all {x1, x2}∈H×H, where {f1, f2}∈S(λ),
{x1−f1, x2−f2}∈N. If f2 =0, then x2∈H0. Hence, {x1, x2}∈N. The uniqueness
of decomposition (26) implies f1 = 0. So, S−1(λ) is the operator.
Let us prove that R(S(λ)) is closed. For this purpose, we consider the ope-
rator U(λ) :R(S(µ))→H defined by the equality U(λ)x = P2P(λ){S−1(µ)x, x},
where x ∈ R(S(µ)), P(λ) is the projection of space H×H onto S(λ) in parallel
N corresponding to (26), P2 is the projection H × H onto the second factor,
i.e., P2{x1, x2} = x2. For fixed λ, the operator U(λ) is a continuous one-to-one
mapping of R(S(µ)) onto R(S(λ)). The operator function U(λ) is holomorphic
in some neighborhood of µ and U(µ)x = x. Therefore the set R(S(λ)) is closed
for all λ from some neighborhood of µ. Lemma 8 is proved.
Corollary 10. Suppose the family of relations S(λ) is holomorphic in a con-
nected domain D, and S(µ) satisfies the assumptions of Lemma 8 for all points
µ ∈ D. Then dimHªR(S(λ)) is constant in D.
Proof follows from the fact that this dimension is constant in a neighborhood
of every point belonging to D.
132 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2
On Linear Relations Generated by a Differential Expression...
4. On the Characteristic Operator
In the space H = L2(H, A(t); a, b), the norm and the scalar product are defined
by equalities (5), (6). According to Statements 1, 3, after replacing A(t) = aλ0(t)
by aλ(t), we obtain a space with the equivalent norm. By ‖·‖aλ
( (·, ·)aλ
) we
denote the norm (the scalar product, respectively) in the space L2(H, aλ(t); a, b).
We consider the equation
l[y]− Bλ(t)y − iAλ(t)y = aλ(t)f(t) (f ∈ H) (27)
on the finite or infinite interval (a, b) and use the following definition (see [4], [5]).
Definition 2. Let M(λ) = M∗(λ̄) be a function holomorphic for Imλ 6= 0
whose values are bounded linear operators such that D(M(λ)) = Q+ and R(M(λ))
⊂ Q−. This function M(λ) is called a characteristic operator of equation (27)
if for any function f ∈ H with compact support the corresponding solution yλ(t)
of (27)
yλ(t) = (Rλf)(t)
=
b∫
a
Wλ(t)(M(λ)− (1/2)sgn(s− t)J−1
r (t0))W ∗̄
λ (s)aλ(s)f(s)ds (28)
satisfies the inequality
‖Rλf‖2
aλ
6 Im(Rλf, f)aλ
/Imλ, Imλ 6= 0. (29)
(For the case of the singular boundary a or b, the spaces Q−, Q+ are defined
bellow.)
In [4], [5], the existence of the characteristic operator is proved. In the case
of the axis, the characteristic operators appear that are unbounded in Q [4,
pp. 167, 172]. In [5, p. 209], there is proposed another method allowing to connect
a holomorphic family of dissipative operators with the expression lλ. (In [4], [5],
the case r = 1 is considered.)
Suppose that the boundaries a, b are regular. Let L0(λ) ⊂ L(λ) ⊂ L(λ) for
Imλ 6= 0. We say that a family of linear relations L(λ) generates a characteristic
operator M(λ) of equation (27) if γ(λ)L(λ) = M−1(λ).
Theorem 8. The family of closed relations L(λ) (Imλ 6=0) generates the cha-
racteristic operator of equation (27) if and only if L(λ) satisfies the following
conditions: 1) L(λ̄) = L∗(λ); 2) the family L(λ) is holomorphic; 3) for fixed λ,
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2 133
V.M. Bruk
the relation L−1(λ) is an everywhere defined operator (L−1(λ) is bounded since
L−1(λ) is closed); 4) the inequality
Im(gλ, zλ)A 6 −(Ã−1
0 Ãλzλ, zλ)A = −
b∫
a
(Ãλ(t)zλ(t), zλ(t))dt (30)
holds for all pairs {zλ, gλ} ∈ L(λ) and all λ such that Imλ > 0.
P r o o f. Assume that the family L(λ) possesses the properties 1)–4). It
follows from Theorems 4, 5 and Lemma 6 that there exists a holomorphic function
M(λ) = M∗(λ̄) (Imλ 6= 0) whose values are the bounded operators such that
D(M(λ)) = Q+, R(M(λ)) ⊂ Q−, and the equality
zλ(t) = (L−1(λ)g)(t)
=
b∫
a
Wλ(t)(M(λ)− (1/2)sgn(s− t)J−1
r (t0))W ∗̄
λ (s)Ã(s)g(s)ds (31)
holds for all g ∈ H. The function M(λ) is defined by the equality
γ(λ)L(λ) = M−1(λ). (32)
We set f(t) = (Imλ)Ã−1
0,λ(t)Ã(t)g(t). It follows from the arguments to the
definition of the spaces H1/2(t), H−1/2(t) and Corollary 3 that the functions f
and g belong or do not belong to the space H together. Moreover, the equalities
g(t) = Ã−1
0 (t)aλ(t)f(t), aλ(t)f(t) = Ã(t)g(t) (33)
hold. We fix the function f ∈ H and denote gλ(t) = Ã−1
0 (t)aλ(t)f(t), yλ =
L−1(λ)gλ = Rλf . By (31), (33), it follows that yλ is the solution of equation (27)
and yλ has the form (28).
Now we claim that inequality (29) holds. Indeed, (30) implies
Im(f,Rλf)aλ
= Im(gλ, yλ)A 6 −Imλ ‖yλ‖2
aλ
= −Imλ ‖Rλf‖2
aλ
for Imλ > 0. If Imλ < 0, then (29) follows from the equality L(λ̄) = L∗(λ).
Thus, if the family L(λ) satisfies the conditions 1)–4), then the function M(λ)
defined by (32) is the characteristic operator. The proof is convertible. From
this there follows the converse. Theorem 8 is proved.
Using Theorems 6, 7, 8, formula (25) and the equality L(λ̄)=L∗(λ), we obtain
Corollary 11. The family M(λ) = (γ(λ)L(λ))−1 is a characteristic operator.
134 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2
On Linear Relations Generated by a Differential Expression...
R e m a r k 4. In (14), suppose that cλ ∈ Q. By Lemma 2, y is the function
ranging in H. Using (14), (4), we get ŷ(a) = Ŵλ(a)cλ, ŷ(b) = Ŵλ(b)cλ + F̂λ(b).
From this and (17), by straightforward calculations, we obtain
Y ′λ =Jr(t0)(Ŵ−1
λ (b)ŷ(b)−Ŵ−1
λ (a)ŷ(a)), Yλ =2−1(Ŵ−1
λ (b)ŷ(b)+Ŵ−1
λ (a)ŷ(a)).
These equalities correspond to Remark 1.1 in [4]. For L(λ), the values ŷ(a), ŷ(b)
are calculated by formula (12), where µ is changed to µ̄.
R e m a r k 5. Let L̃(µ0) be the maximal relation and L̃0(µ0) be the minimal
relation generated by the formally self-adjoint expression l[y]−Bµ0(t)y in H, where
µ0 ∈ C0 is fixed. It follows from Theorem 8 that the family L(λ) generates a
characteristic operator if and only if the family Λ(λ) = L(λ) + B̃λ,µ0 + iÃλ has
the properties: 1) L̃0(µ0) ⊂ Λ(λ) ⊂ L̃(µ0); 2) Λ∗(λ) = Λ(λ̄); 3) the family
Λ(λ) is holomorphic for Imλ 6= 0; 4) Λ(λ) is a maximal accumulative relation
for Imλ > 0. Thus, (Λ(λ) − λE)−1 is a generalized resolvent of L̃0(µ0). This
statement is established by the other method in [4].
Below we assume that a = −∞, b = ∞. So, H = L2(H, A(t);−∞,∞). We
define maximal and minimal relations in the following way. Let D′
0,λ (λ ∈ C0) be a
set of finite functions y ∈ H satisfying the conditions: (a) the quasiderivatives y[k]
exist, they are absolutely continuous up to the order r−1 inclusively; (b) lλ[y](t) ∈
H1/2(t) almost everywhere; (c) Ã−1
0 (t)lλ[y] ∈ H. We introduce a correspondence
between each class of functions identified with y ∈ D′
0,λ in H and the class of
functions identified with Ã−1
0 (t)lλ[y] in H. Thus, in the space H, we obtain a
linear relation L′0(λ), denote its closure by L0(λ), and call L0(λ) the minimal
relation. (In the regular case, one can define the minimal relation in the same
way. One can show that this definition and the definition of minimal relation from
Section 2 are equivalent. We do not use this fact.) We denote L(λ) = (L0(λ̄))∗.
The relation L(λ) is called maximal. (According to Lemma 4, in the regular case
the maximal relation can be defined in the same way.)
Using the equality L′0(µ) = L′0(λ) + C̃λ,µ (λ, µ ∈ C0) and Corollary 2, we
get L0(µ) = L0(λ) + C̃λ,µ. This implies that the family L0(λ) is holomorphic on
C0. By Lemma 7, it follows that the family L(λ) is holomorphic on C0. Using
Corollary 2, we get L(µ) = L(λ) + C̃λ,µ. Thus, Theorem 1 and Corollary 7 are
valid for the relations L(λ) and L0(λ). For the relation L′0(λ) (Imλ 6= 0), the
proof of Lemma 5 is the same as the above proof for the regular case. By the limit
passage, we obtain that Lemma 5 is valid for relation L0(λ). Consequently, the
range R(L0(λ)) is closed in H and the relation L−1
0 (λ) is a bounded everywhere
defined operator on R(L0(λ)) (Imλ 6= 0).
In Corollary 10, we set S(λ) = L0(λ). Then we obtain that the dimension
dimHªR(L0(λ)) = dimkerL(λ̄) is constant in the half-planes Imλ > 0, Imλ < 0.
In [4, 6, 7, 21–23] this assertion was proved by various methods for various cases.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2 135
V.M. Bruk
We fix some µ (Imµ > 0) and denote Nµ = HªR(L0(µ)) (it is not inconceiv-
able that Nµ = {0}). Obviously, Nµ = ker(L0(µ))∗ = kerL(µ̄). The definition
of the relation L(λ) is the same as the above for the regular case, i.e, L(λ) is
the restriction L(λ) to D(L0(λ)) u Nµ. The relation L(λ) consists of the ordered
pairs of the form (24). We set L(λ) = L∗(λ̄) for Imλ < 0. Theorems 6, 7 are
obtained by the literal repetition of the proof used for the regular case.
We construct the domain and the range of the characteristic operator. By Q0
denote a set of elements c ∈ Hr such that the norm ‖A(t)Wµ(t)c‖ = 0 almost
everywhere on (−∞,∞), Q = HrªQ0. The sets Q0, Q do not depend on µ ∈ C0.
(This assertion can be proved by analogy with the corresponding assertion for
the regular case.)
Let [αn, βn] be a sequence of intervals such that [αn, βn]⊂ (αn+1, βn+1), and
αn→−∞, βn→∞ as n→∞ (n∈N). In Q, we introduce the system of seminorms
pn(x) =
βn∫
αn
∥∥∥A1/2(s)Wµ(s)x
∥∥∥
2
ds
1/2
, µ ∈ C0, x ∈ Q. (34)
We denote the completion of Q with respect to this system by Q−. The
space Q− is locally convex. Arguing as above, we see that the replacement
of µ by λ ∈ C0 leads to an isomorphic space [25, Ch. 1]. The space Q− is
isomorphic to a projective limit of a family of Banach spaces Q−(n) (see [25,
Ch. 2, proof 5.4]). The spaces Q−(n) are constructed bellow. Let Q+(n) be
the space adjoint to Q−(n), Q+ be an inductive limit of the spaces Q+(n). We
claim that D(M(λ))=Q+, R(M(λ))⊂Q−. We describe this construction more
detail for the proof of properties of the operator M(λ). This construction is little
distinguished from the corresponding construction in [24]. Therefore we omit the
details of argumentation.
We denote Hn = L2(H, A(t);αn, βn), n ∈ N. Let Q0(n) be a set of elements
c ∈ Q such that the function Wλ(t)c is identified with zero in the space Hn,
Q(n)=Q ª Q0(n). Then Q0(n)⊂Q0(m) and Q(n)⊃Q(m) for n>m. In Q(n),
we introduce the seminorm pn(x) by formula (34). This seminorm is the norm
on Q(n). We denote it by ‖·‖(n)
− . If m < n, c ∈ Q(m), then ‖c‖(m)
− 6 ‖c‖(n)
− .
Let Q−(n) be the completion of Q(n) with respect to the norm ‖·‖(n)
− . We define
mappings hmn : Q−(n)→Q−(m) (n > m) in the following way. Let Q
(n)
− (n,m)
be the completion of Q(n,m)=Q(n)ªQ(m) with respect to the norm ‖·‖(n)
− and
Q
(n)
− (m) be the completion of Q(m) with respect to this norm. Then Q−(n) =
Q
(n)
− (m)uQ
(n)
− (n,m). We define hm,nc=0 whenever c∈Q
(n)
− (n,m), and we define
hm,nc= jmnc whenever c ∈ Q
(n)
− (m), where jmn is the inclusion map of Q
(n)
− (m)
into Q−(m). We set hnnc=c for m=n. The mappings hmn are continuous.
136 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2
On Linear Relations Generated by a Differential Expression...
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) [25, Ch. 2].
By repeating the corresponding arguments from [25, Ch. 2], one can show that
this projective limit is isomorphic to the space Q− introduced after formula (34).
We do not use this fact and thereby omit its proof. Further, by Q− we denote
the projective limit lim(pr)hmnQ−(n). It follows from the definition of projective
limit [25, Ch. 2] that Q− is a subspace 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 a treated as a negative one with respect to Q(n)
[8, Ch. 2]. By Q+(n) we 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+ we denote the inductive limit [25, 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 [25, 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 [25, Ch. 2].
By Lemma 2, the operator cn→Wλ(t)cn is a continuous one-to-one mapping
of Q−(n) into Hn and it has the closed range. Denote this operator by Wλ(n).
It follows from Lemma 2 that the adjoint operator W∗
λ(n) maps continuously
Hn onto Q+(n), and W∗
λ(n)f =
∫ βn
αn
W ∗
λ (s)Ã(s)f(s)ds. Hence, for each function
f ∈ H and all finite α, β, we have
∫ β
α W ∗
λ (s)Ã(s)f(s)ds ∈ Q+.
Let c={cn}∈Q−. Then cm =hmncn (m 6 n). Consequently, the restriction of
the function Ã1/2(t)Wλ(t)cn to the segment [αm, βm] coincides with the function
Ã1/2(t)Wλ(t)cm in the space L2(H;αm, βm). By Ã1/2(t)Wλ(t)c we denote the
function that is equal to Ã1/2(t)Wλ(t)cn on each segment [αn, βn]. Now by Wλ(t)c
denote the function ranging in H−1/2(t) ⊕ G(t) and coinciding with Wλ(t)cn in
the space Hn for each n ∈ N. So, for all m, n (m 6 n), the equality Wλ(t)cn =
Wλ(t)cm holds in the space Hm.
Lemma 9. If the pair {y, f} ∈ L(λ), then y has the form
y(t) = Wλ(t)c + Wλ(t)J−1
r (t0)
t∫
t0
W ∗̄
λ (s)Ã(s)f(s)ds,
where c ∈ Q−.
P r o o f follows from Lemma 3 and the definition of the function Wλ(t)c.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 2 137
V.M. Bruk
Theorem 9. For fixed λ (Imλ 6= 0), the operator L−1(λ) is integral
L−1(λ)g =
∞∫
−∞
Kλ(t, s)Ã(s)g(s)ds, g ∈ H,
where Kλ(t, s)=Wλ(t)(M(λ)−(1/2)sgn(s−t)J−1
r (t0))W ∗̄
λ
(s) and M(λ) :Q+→Q−
is a continuous operator such that M(λ̄) = M∗(λ). The operator function M(λ)x
is holomorphic for Imλ 6= 0 and each x ∈ Q+.
R e m a r k 6. In Theorem 9, the integral converges at least weakly in H.
P r o o f. As established above, the relation L−1(λ) is a bounded everywhere
defined operator in the space H and the operator function L−1(λ) is holomorphic
for Imλ 6= 0. By this fact, the further proof of the theorem and the remark repeats
word-for-word the proof of the main theorem in [24]. Theorem 9 is proved.
Theorem 10. In Theorem 9, the operator function M(λ) is the characteristic
operator of equation (27) on the axis (−∞,∞).
P r o o f. It follows from Theorem 9 that for any finite function g ∈ H the
equality (31) holds with the replacement of a by −∞ and of b by +∞. The
family of relations L(λ) satisfies all conditions of Theorem 8. We obtain the
desired assertion by repeating the proof of this theorem. Theorem 10 is proved.
The author wishes to express gratitude to the referee for useful remarks.
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