Generalized Resolvents of Symmetric Relations Generated on Semi-Axis by a Differential Expression and a Nonnegative Operator Function
Generalized resolvents of a minimal symmetric relation generated on the semi-axis by a formally selfadjoint di erential expression and a nonnegative operator function are described.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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| Назва видання: | Журнал математической физики, анализа, геометрии |
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| Цитувати: | Generalized Resolvents of Symmetric Relations Generated on Semi-Axis by a Differential Expression and a Nonnegative Operator Function / V.M. Bruk // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 372-387. — Бібліогр.: 11 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1066762016-10-03T03:02:12Z Generalized Resolvents of Symmetric Relations Generated on Semi-Axis by a Differential Expression and a Nonnegative Operator Function Bruk, V.M. Generalized resolvents of a minimal symmetric relation generated on the semi-axis by a formally selfadjoint di erential expression and a nonnegative operator function are described. 2006 Article Generalized Resolvents of Symmetric Relations Generated on Semi-Axis by a Differential Expression and a Nonnegative Operator Function / V.M. Bruk // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 372-387. — Бібліогр.: 11 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106676 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
Generalized resolvents of a minimal symmetric relation generated on the semi-axis by a formally selfadjoint di erential expression and a nonnegative operator function are described. |
| format |
Article |
| author |
Bruk, V.M. |
| spellingShingle |
Bruk, V.M. Generalized Resolvents of Symmetric Relations Generated on Semi-Axis by a Differential Expression and a Nonnegative Operator Function Журнал математической физики, анализа, геометрии |
| author_facet |
Bruk, V.M. |
| author_sort |
Bruk, V.M. |
| title |
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis by a Differential Expression and a Nonnegative Operator Function |
| title_short |
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis by a Differential Expression and a Nonnegative Operator Function |
| title_full |
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis by a Differential Expression and a Nonnegative Operator Function |
| title_fullStr |
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis by a Differential Expression and a Nonnegative Operator Function |
| title_full_unstemmed |
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis by a Differential Expression and a Nonnegative Operator Function |
| title_sort |
generalized resolvents of symmetric relations generated on semi-axis by a differential expression and a nonnegative operator function |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106676 |
| citation_txt |
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis by a Differential Expression and a Nonnegative Operator Function / V.M. Bruk // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 372-387. — Бібліогр.: 11 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT brukvm generalizedresolventsofsymmetricrelationsgeneratedonsemiaxisbyadifferentialexpressionandanonnegativeoperatorfunction |
| first_indexed |
2025-07-07T18:51:07Z |
| last_indexed |
2025-07-07T18:51:07Z |
| _version_ |
1837015256121999360 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 4, pp. 372�387
Generalized Resolvents of Symmetric Relations
Generated on Semi-Axis by a Di�erential Expression and
a Nonnegative Operator Function
V.M. Bruk
Saratov State Technical University
77 Politechnitseskaja Str., Saratov, 410054, Russia
E-mail:bruk@san.ru
Received July 10, 2005
Generalized resolvents of a minimal symmetric relation generated on the
semi-axis by a formally selfadjoint di�erential expression and a nonnegative
operator function are described.
Key words: symmetric relation, generalized resolvent, characteristic ope-
rator function, inductive limit, projective limit.
Mathematics Subject Classi�cation 2000: 47A06, 47A10, 34B27.
1. Introduction
In [1], A.V. Straus described the generalized resolvents of the symmetric ope-
rator generated by a formally selfadjoint di�erential expression of even order in
a scalar case. In [2] these results were used for the operator case. A di�erential
expression with a nonnegative weight generates a linear relation. This relation
is not an operator, in general. The generalized resolvents formulae for these
relations are given in [3�5]. However, in these papers either the �nite-dimensional
case [3, 5] or the in�nite-dimensional case [3, 4] under conditions that the kernel
(the null space) of the maximal relation contained only solutions of the corres-
ponding homogeneous equation was considered. In our paper a general situation is
considered. We use projective and inductive limits of special spaces in the singular
case to construct the spaces where a characteristic operator function acts. We
consider the case of semi-axis instead of the general singular case only to simplify
notations. The detailed bibliography is given in [1�5] and in the monograph [6].
c
V.M. Bruk, 2006
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis...
2. Notations and Auxiliary Formulae
Let H be a separable Hilbert space with the scalar product (�; �) and the norm
k�k; A(t) be an operator function strongly measurable on the interval [a;1);
the values of A(t) are bounded operators in H such that for all x 2 H the scalar
product (A(t)x; x) � 0 almost everywhere. Suppose the norm kA(t)k is integrable
on every compact interval [a; �] � [a;1).
We denote by l the di�erential expression of order r (r = 2n or r = 2n+ 1):
l[y] =
8>><
>>:
nP
k=1
(�1)kf(pn�k(t)y
(k))(k) � i[(qn�k(t)y
(k))(k�1) + (qn�k(t)y
(k�1))(k)]g+ pn(t)y;
nP
k=0
(�1)kfi[(qn�k(t)y
(k))(k+1) + (qn�k(t)y
(k+1))(k)] + (pn�k(t)y
(k))(k)g:
Coe�cients of l are bounded selfadjoint operators in H. The leading coe�cients,
p0(t) in the case of r = 2n and q0(t) in the case of r = 2n+ 1, have the bounded
inverse operator almost everywhere. The functions pn�k(t) are strongly di�eren-
tiable k times and the functions qn�k(t) are strongly di�erentiable k times in the
case r = 2n, and k+1 times in the case r = 2n+1. In general, we do not assume
the coe�cients of the expression l to be smooth as we have just said. According to
[7] we treat l as a quasidi�erential expression. Quasi-derivatives for the expression
l are de�ned in [7]. Suppose the functions pj(t), qm(t) are strongly measurable,
the function q0(t) in the case r = 2n+1 is strongly di�erentiable, and the norms
of functions
p�10 (t); p�10 (t)q0(t); q0(t)p
�1
0 (t)q0(t); p1(t); : : : ; pn(t); q0(t); : : : ; qn�1(t)
(in the case r = 2n );
q00(t); q1(t); : : : ; qn(t); p0(t); : : : ; pn(t)
(in the case r = 2n+ 1)
are integrable on every compact interval [a; �] � [a;1).
We de�ne the scalar product
hy1; y2i =
1Z
a
(A(t)y1(t); y2(t))dt;
where yi(t) are H-valued functions continuous on [a;1), and
1R
a
A1=2(t)yi(t)
2 dt
< 1, i = 1; 2. By identifying with zero the functions y such that hy; yi = 0
and making the completion, we obtain the Hilbert space. We denote this space
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 373
V.M. Bruk
by B = L2(H;A(t); a;1). Let ~y be some element belonging to B, i.e., ~y is
a corresponding class of functions. If y1; y2 2 ~y, then y1; y2 are identi�ed with
respect to the norm generated by the scalar product h�; �i. By ~y we denote the
class of functions containing y. Suppose y 2 ~y. Without loss of generality, further
we will often say that y(t) belongs to B.
Let (a0; b0) � [a;1) and B0 = L2(H;A(t); a0; b0). If ~y 2 B0, then extending
y by zero to the whole interval [a;1) we can consider that ~y 2 B. If ~y 2 B, then
restricting y to the interval (a0; b0) we can consider that ~y 2 B0 (it is not excepted
that ~y 6= 0 in B and ~y = 0 in B0).
Let G(t) be the set of elements x 2 H such that A(t)x = 0, and H(t) be the
orthogonal complement of G(t) in H, H = H(t)�G(t), and A0(t) be the restric-
tion of A(t) to H(t). Suppose H� (t), �1 < � <1, is the Hilbert scale of spaces
[8, Ch. 2] generated by the operator A�10 (t). For the �xed t; operator A
1=2
0 (t)
is a continuous one-to-one mapping of H(t) = H0(t) onto H1=2(t). We denote
the adjoint operator of A
1=2
0 (t) by Â
1=2
0 (t). The operator Â
1=2
0 (t) is a continuous
one-to-one mapping of H
�1=2(t) onto H(t) and Â
1=2
0 (t) is an extension of A
1=2
0 (t).
Let ~A0(t) = A
1=2
0 (t)Â
1=2
0 (t). The operator ~A0(t) is a continuous one-to-one map-
ping of H
�1=2(t) onto H1=2(t) and ~A0(t) is an extension of A0(t). We denote
~A(t) (respectively ~A1=2(t)) the operator de�ned on H
�1=2 �G(t) such that ~A(t)
( ~A1=2(t)) is equal to ~A0(t) (respectively Â
1=2
0 (t)) on H
�1=2(t) and ~A(t)( ~A1=2(t)) is
equal to zero on G(t). The operator ~A(t) ( ~A1=2(t)) is an extension of A(t) (A1=2(t)
respectively).
In [3] it is proved that spaces H
�1=2(t) are measurable with respect to pa-
rameter t [9, Ch. 1] whenever we take functions of the form ~A�10 (t)A1=2(t)h(t)
instead of measurable functions, where h(t) is a measurable H-valued function.
The space B is a measurable sum of spaces H
�1=2(t) and B consists of elements
(i.e., classes of functions) with representatives of the form ~A�10 (t)A1=2(t)h(t),
where h(t) 2 L2(H; a;1), i.e.,
1R
a
kh(t)k2 dt < 1. If y1, y2 are representatives
of the class of functions ~y 2 B, then ~A1=2(t)y1(t), ~A1=2(t)y2(t) are the same func-
tions in the space L2(H; a;1). We denote this function by ~A1=2(t)~y.
We de�ne minimal and maximal relations generated by the expression l and
the function A(t) in the following way. Let D0
0 be the set of �nite on (a;1) func-
tions y satisfying the following conditions: a) the quasi-derivatives y[0]; : : : ; y[r]
of function y exist, they are absolutely continuous up to the order r � 1; b)
l[y](t) 2 H1=2(t) almost everywhere; c) the function ~A�10 (t)l[y] belongs to B. To
each class of functions identi�ed in B with y 2 D0
0 we assign the class of functions
identi�ed in B with ~A�10 (t)l[y]. This correspondence L00 may not be an operator
as it may happen that some function y is identi�ed with zero in B and ~A�10 (t)l[y]
374 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis...
is not equal to zero. So, we get a linear relation L00 in the space B. The closure
of L00 we denote by L0. The relation L0 is called as a minimal one. Let L�0 be the
relation adjoint of L0. L
�
0 is called the maximal relation.
Terminology concerning linear relations can be found in the monographs [6,
8]. Further the following notations are used: R as a range of values; f�; �g as an
ordered pair.
We consider the di�erential equation l[y] = �A(t)y, where � is a complex
number. Let Wj(t; �) be the operator solution of this equation satisfying the
initial conditions: W
[k�1]
j
(a; �) = ÆjkE (E is the identity operator, Æjk is the
Kronecker symbol, j; k = 1; : : : ; r). By W (t; �) we denote the one-row operator
matrix (W1(t; �); : : : ;Wr(t; �)). The operator W (t; �) maps continuously Hr into
H for �xed t, �. The adjoint operator W �(t; �) maps continuously H into Hr.
If l[y] exists for the function y, then we denote ŷ = (y; y[1]; : : : ; y[r�1]) (we treat
ŷ as a one-columned matrix). Let z = (z1; : : : ; zm) be some system of functions
such that l[zj ] exists for each j. By ẑ we denote the matrix (ẑ1; : : : ; ẑm). The
analogous notations are used for the operator functions.
We consider the operator matrices of orders 2n and 2n+1 for the expression l
in cases r = 2n and r = 2n+ 1 respectively:
J2n(t) =
0
BBBBBB@
�E
:::
�E
E
:::
E
1
CCCCCCA
;
J2n+1(t) =
0
BBBBBBBB@
�E
:::
�E
2iq�10 (t)
E
:::
E
1
CCCCCCCCA
;
where all the elements, that are not indicated, are equal to zero. (In matrix
J2n+1(t) the element 2iq�10 (t) stands on the intersection of the row n+1 and the
column n+1.) Suppose the expression l is de�ned for the functions y, z, then, in
these notations, Lagrange's formula has the following form:
�Z
�
(l[y]; z)dt �
�Z
�
(y; l[z])dt = (Jr(t)ŷ(t); ẑ(t))j
�
�
; a � � < � <1: (1)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 375
V.M. Bruk
It follows from "method of the variation of arbitrary constants" that general
solution of the equation
l[y]� � ~A(t)y = ~A(t)f(t)
is represented in the form:
y(t) = W (t; �)
0
@c+ J�1r (a)
tZ
a
W �(s; ��) ~A(s)f(s)ds
1
A ; (2)
where c 2 Hr. Consequently,
ŷ(t) = Ŵ (t; �)
0
@c+ J�1r (a)
tZ
a
W �(s; ��) ~A(s)f(s)ds
1
A : (3)
3. Construction of a Space Containing the Range
of the Characteristic Operator Function M(�)
Let Q0 be a set of elements c 2 Hr such that function W (t; 0)c is identi�ed
with zero in the space B, i.e.,
1R
a
A1=2(s)W (s; 0)c
2 ds = 0. It follows from the
equalities
W (t; �)c = W (t; 0)
0
@c+ �J�1r (a)
tZ
a
W �(s; 0) ~A(s)W (s; �)cds
1
A ; (4)
W (t; 0)c = W (t; �)
0
@c� �J�1r (a)
tZ
a
W �(s; ��) ~A(s)W (s; 0)cds
1
A (5)
that the function W (t; �)c is identi�ed with zero in the space B if and only if
c 2 Q0 (in the �nite-dimensional case this fact was obtained in [7]). By Q we
denote an orthogonal complement of Q0 in Hr, Hr = Q�Q0.
Let [a; �m],m = 1; 2; : : : , be a system of intervals such that [a; �m] � [a; �m+1)
and �m ! 1 as m ! 1. We denote Bm = L2(H;A(t); a; �m). Suppose Q0(m)
is the set of elements c 2 Q such that the function W (t; �)c is identi�ed with zero
in the space Bm, i.e.,
�mR
a
A1=2(s)W (s; �)c
2 ds = 0. It follows from (4), (5) that
Q0(m) does not depend on �. Let Q(m) be the orthogonal complement of Q0(m)
376 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis...
in Q, i.e., Q = Q(m) �Q0(m). Obviously, Q0(1) � Q0(2) � : : : � Q0(m) � : : :
and Q(1) � Q(2) � : : : � Q(m) � : : : � Q.
We de�ne the quasiscalar product
(c; d)
(i)
�
=
�iZ
a
( ~A(s)W (s; 0)c;W (s; 0)d)ds; c; d 2 Q;
in space Q. This quasiscalar product generates the semi-norm
kck
(i)
�
=
0
@
�iZ
a
A1=2(s)W (s; 0)c
2 ds
1
A
1=2
�
kck ; c 2 Q;
=
(i) > 0: (6)
Clearly, k�ki
�
� k�k
i+1
�
.
Note that if c 2 Q(m), then kck
(m)
�
> 0 for c 6= 0. Therefore the semi-norm
k�k
i
�
is a norm on the set Q(m) for i � m. By Q
(i)
�
(m) we denote the completion of
Q(m) with respect to this norm. It follows from (4), (5) that we obtain the same
set Q
(i)
�
(m) with the equivalent norm whenever we replace W (s; 0) by W (s; �)
in (6). The inclusion map Q
(k)
�
(m) � Q
(i)
�
(m) is continuous for k � i � m. We
denote Q
�
(m) = Q
(m)
�
(m).
Let ker(a; �m; �) be a closure of the set of elements (i.e., of classes of functions)
in the space Bm with the representatives of the form W (t; �)x, where x 2 Q(m).
(We denote these classes by ~W (t; �)x.) It follows from (4�6) that the operator
c! ~W (t; �)c (c 2 Q
�
(m)) is the continuous one-to-one mapping of Q
�
(m) onto
ker(a; �m; �). By Wm(�) we denote this operator. Here ~W (t; �)c is the class of
functions such that the sequence f ~W (t; �)ckg converges to ~W (t; �)c in the space
Bm whenever fckg converges to c in the space Q
�
(m).
By Q(n;m) we denote the orthogonal complement of Q(m) in Q(n) for n > m,
i.e., Q(n) = Q(m)�Q(n;m). Then
Q
�
(n) = Q
(n)
�
(m) _+Q
(n)
�
(n;m); (7)
where Q
(n)
�
(n;m) is the completion of Q(n;m) with respect to the norm k�k
(n)
�
.
Hence denoting
ker(Q(m); a; �n; �) = Wn(�)Q
(n)
�
(m); ker(Q(n;m); a; �n; �) = Wn(�)Q
(n)
�
(n;m);
we obtain
ker(a; �n; �) = ker(Q(m); a; �n; �) _+ ker(Q(n;m); a; �n; �): (8)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 377
V.M. Bruk
We de�ne the linear mappings gmn : ker(a; �n; �) ! ker(a; �m; �) (n � m) in
the following way. Let gmnz = Wm(�)jmnW
�1
n (�)z for z 2 ker(Q(m); a; �n; �)
and gmnz = 0 for z 2 ker(Q(n;m); a; �n; �) (here jmn is the inclusion map of
Q
(n)
�
(m) into Q
�
(m)). It follows from (7), (8) and the properties of the operators
Wk(�) that mappings gmn are continuous.
Moreover, we introduce the linear mappings hmn : Q
�
(n) ! Q
�
(m) (n � m)
in accordance with (7) in the following way. Since the inclusion map of Q
(n)
�
(m)
into Q
�
(m) is continuous, we de�ne hmnc = jmnc whenever c 2 Q
(n)
�
(m), and we
de�ne hmnc = 0 whenever c 2 Q
(n)
�
(n;m). Mappings hmn are continuous.
By ker(a;1; �) we denote a projective limit of the family fker(a; �n; �); n 2Ng
with respect to mappings gmn and by Q� we denote a projective limit of the family
fQ
�
(n));n 2Ng with respect to mappings hmn, i.e.,
ker(a;1; �) = lim(pr)gmn ker(a; �n; �); Q
�
= lim(pr)hmnQ�(n):
It follows from the de�nition of projective limit [10, Ch. 2] that Q
�
is a sub-
space of the product
Q
n
Q
�
(n) and Q
�
consists of the elements c = fcng such that
cm = hmncn for all m � n. Similarly, ker(a;1; �) is a subspace of
Q
n
ker(a; �n; �)
and the analogous statement is true in regard to ker(a;1; �). By pn, p
0
n we de-
note the projections
Q
n
Q
�
(n) and
Q
n
ker(a; �n; �) onto Q
�
(n) and ker(a; �n; �)
respectively.
The mappings gmn, hmn and the operators Wn(�) : Q
�
(n) ! ker(a; �n; �)
satisfy the equality: gmn = Wm(�)hmnW
�1
n (�). Consequently, the family of
operators fWn(�)g generates the isomorphism (i.e., the linear homeomorphism)
W (�) : Q
�
! ker(a;1; �). If c = fcng 2
Q
n
Q
�
(n), then W (�)c = fWn(�)cng
and W (�)Q
�
= ker(a;1; �). Moreover,
p0n(W (�)Q
�
) = Wn(�)pn(Q�): (9)
Lemma 1. Let wn be a representative of the class of functions ~wn = Wn(�)d
(d 2 Q
�
(n)) and let wm be restriction of wn to [a; �m] (m � n). Then
~wm = Wm(�)hmnd: (10)
P r o o f. According to (7) we represent d in the form d = d0 + d00, where
d0 2 Q
(n)
�
(m) � Q
�
(m), d00 2 Q
(n)
�
(n;m). Suppose the sequences fd0
k
g, fd00
k
g
(d0
k
2 Q(m), d00
k
2 Q(n;m)) converge to d0, d00 in the spaces Q
(n)
�
(m), Q
(n)
�
(n;m)
respectively. Then the sequence f ~W (t; �)d0
k
g converges to Wn(�)d
0 in the space
Bn. Therefore f ~W (t; �)d0
k
g converges to Wm(�)jmnd
0 in Bm and the functions
W (t; �)d00
k
are identi�ed with zero in Bm. Hence follows (10). Lemma 1 is proved.
378 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis...
Let c0 2 Hr. Then the function ~A1=2(t)W (t; �)c0 belongs to L2(H; a; �n) for
all n and ~A1=2(t)W (t; �)c0 coincides with ~A1=2(t)W (t; �)c00 in this space, where
c00 = PnP0c 2 Q(n), P0 is the orthogonal projection of Hr onto Q, Pn is the
orthogonal projection of Q onto Q(n). Suppose the sequence fdkg, dk 2 Q(n),
converges to d in the space Q
�
(n); then classes of functions ~W (t; �)dk 2 Bn
with the representatives of W (t; �)dk converge to the class of functions ~W (t; �)d
in Bn. Therefore functions ~A1=2(t)W (t; �)dk converge to the function z(t) =
~A1=2(t) ~W (t; �)d in the space L2(H; a; �n). It follows from (10) that the restriction
of z(t) to the interval [a; �m], m < n, coincides with ~A1=2(t) ~W (t; �)hmnd.
Suppose c = fcng 2 Q
�
; then cm = hmncn (m � n). It follows from (10)
that the restriction of function ~A1=2(t) ~W (t; �)cn to the interval [a; �m] coin-
cides with the function ~A1=2(t) ~W (t; �)cm in the space L2(H; a; �m). Therefore
by ~A1=2(t) ~W (t; �)c we denote the function coinciding with ~A1=2(t) ~W (t; �)cn on
any interval [a; �n]. Correspondingly, by ~W (t; �)c we denote the H
�1=2(t)�G(t)-
valued function coinciding with ~W (t; �)cn in the spaces Bn for all n. It follows
from (9), (10) that ~W (t; �)cn = ~W (t; �)cm in the space Bm, m � n.
4. Construction of a Domain of the Characteristic Operator
Function M(�)
The space Q
�
(n) can be treated as a negative one with respect to Q(n).
By Q+(n) we denote a corresponding space with the positive norm. It follows
from (7) that Q+(n) = Q
(n)
+ (m) _+Q
(n)
+ (n;m), where Q
(n)
+ (m), Q
(n)
+ (n;m) are the
corresponding positive spaces with respect to Q
(n)
�
(m), Q(m) and Q
(n)
�
(n;m),
Q(n;m). The inclusion Q+(m) � Q
(n)
+ (m) is dense and continuous. Consequently
the inclusion map of Q+(m) into Q+(n) is continuous for m � n.
Suppose h+nm : Q+(m) ! Q+(n), n � m, is the adjoint mapping of hmn; then
h+nm is the continuous inclusion map of Q+(m) into Q+(n). By Q+ we denote
inductive limit [10, Ch. 2] of the family fQ+(n);n 2 Ng with respect to mappings
h+nm, i.e., Q+ = lim(ind)h+nmQ+(n). It follows from [10, Ch. 4] that Q+ is the
adjoint space of Q
�
. The space Q+ can be treated as the union Q+ =
S
n
Q+(n)
with the strongest topology such that all inclusion maps of Q+(n) into Q+ are
continuous [10, Ch. 2].
Let ~y 2 Bm and m � n. Suppose y is a representative of the class of func-
tions ~y, then we can treat ~y as an element of the space Bn whenever we extend y
by zero out of the interval [a; �m]. If m � n, then the space Bm can be treated
as a subspace Bn. The topology of Bm is induced by the topology of Bn. Let
inm be the inclusion map of Bm into Bn. By ~B we denote the inductive limit of
the spaces Bn with respect to the mappings inm, i.e., ~B = lim(ind)inmBn. The
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 379
V.M. Bruk
space ~B can be treated as ~B =
S
n
Bn with the strongest topology such that all
inclusion maps of Bn into ~B are continuous.
Suppose fFng, n 2 N, is a family of locally convex spaces such that Fm �
Fn for m � n and this inclusion map is continuous. According to [8, Ch. 1],
an inductive limit F = lim(ind)Fn of the locally convex spaces Fn, n 2 N, is
called a regular one if for every bounded set S � F there is n 2 N such that
S � Fn and S is a bounded set in Fn. It follows from [8, Ch. 1] that the inductive
limits Q+ and ~B are regular. According to [10, Ch. 2], the inductive limit of
bornological spaces is a bornological space. Since Q+, ~B are the inductive limits
of the re�exive Banach spaces, we see that Q+, ~B are bornological.
Suppose Fn are bornological spaces such that their inductive limit F is regular.
Let F0 be a locally convex space. It follows from [10, Ch. 2] that a linear mapping
u : F ! F0 is continuous if and only if for every n 2 N restriction of u to Fn
maps every bounded set S � Fn into the bounded set u(S) � F0. According to
[10, Ch. 2, Ex. 17], we can take a bounded sequence instead of the bounded set
S � Fn. Further, these statements will be used for the proof of the continuity
of corresponding operators. We take the space Q
�
instead of F0. Then the
following conditions are equivalent: (i) the set u(S) is bounded in Q
�
; (ii) the
sets pku(S) are bounded in the spaces pkQ� = Q
�
(k) for every k 2 N; (iii) the
sets Wk(�)pku(S) are bounded in the spaces Bk for every k 2 N; (iiii) the sets
consisting of elements of the form ~W (t; �)ck are bounded for every k 2 N, where
ck 2 pku(S) � Q
�
(k).
Thus the following lemma is proved.
Lemma 2. Suppose the spaces Fn are bornological and their inductive limit F
is regular. The linear operator u : F ! Q
�
is continuous if and only if for every
n 2 N and every bounded set S � Fn and every k 2 N the sets consisting of
elements of the form ~W (t; �)ck are bounded in Bk, where ck 2 pku(S) � Q
�
(k).
Any bounded sequence can be taken in place of bounded set S.
Further, we shall take a family of space fQ+(n)g or fBng in place of fFng.
Then F = Q+ or F = ~B respectively. As it was mentioned above, the operator
Wn(�) is a continuous one-to-one mapping of Q
�
(n) onto the closed subspace
ker(a; �n; �) of the space Bn. Then the adjoint operatorW
�
n(�) maps continuously
Bn onto Q+(n). Therefore W
�
n(�) is the continuous operator of Bn into Q+. The
operator W �
n(�) has the following form:
W �
n(�)
~f =
�nZ
a
W �(s; �) ~A(s)f(s)ds =
1Z
a
W �(s; �) ~A(s)f(s)ds; (11)
where ~f 2 B and f vanishes outside [a; �n].
380 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis...
We note that the norms
W �(s; �)A1=2(s)
,
A1=2(s)f(s)
belong to L2(a; �n).
Hence the integral in the right side of (11) exists.
Since ~B consists of �nite functions, in accordance with (11) we can de�ne the
operator W
�
(�) mapping ~B onto Q+ by the formula
W
�
(�) ~f =
1Z
a
W �(s; ��) ~A(s)f(s)ds:
It follows from the reasoning given before Lemma 2 that the operator W
�
(�) :
~B ! Q+ is continuous. Obviously, W
�
(�) ~f = W �
n(
��) ~f for ~f 2 Bn.
5. The Main Result
To prove the main theorem we need several lemmas.
Lemma 3. ~g 2 B belongs to the range R(L00�
��E) of the relation L00�
��E if
and only if there is an interval (a; �n) such that g is �nite on (a; �n) and
�nZ
a
W �(s; �) ~A(s)g(s)ds = 0: (12)
P r o o f. Let g be �nite and (12) is true. We denote
z(t) = W (t; ��)
0
@J�1r (a)
tZ
a
W �(s; �) ~A(s)g(s)ds
1
A :
From (2), (3), (12) we obtain that the ordered pair f~z; ~gg 2 L00 �
��E.
Vice versa, let f~z; ~gg 2 L00 �
��E. It follows from (2), (3) that there is a rep-
resentative z of the class of functions ~z such that the equality
ẑ(t) = Ŵ (t; ��)
0
@c+ J�1r (a)
tZ
a
W �(s; �) ~A(s)g(s)ds
1
A
is true, where c 2 Q. Since the function z is �nite, we see that c = 0 and g is
�nite on some interval (a; �n) and equality (12) is true. Lemma 3 is proved.
R e m a r k. In Lemma 3 we can replace the interval (a; �n) by any interval
such that the function z vanishes out of this interval, where f~z; ~gg 2 L00 �
��E.
Equality (12) and the equality (~g; ~W (t; �)c)Bn
= 0 are equivalent for all c 2
Q
�
(n).
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 381
V.M. Bruk
Lemma 4. If the ordered pair f~y; ~fg 2 L�0��E, then ~y can be represented in
the following form:
~y(t) = ~W (t; �)c + ~W (t; �)J�1r (a)
tZ
a
W �(s; ��) ~A(s)f(s)ds
= ~W (t; �)
0
@c+ J�1r (a)
tZ
a
W �(s; ��) ~A(s)f(s)ds
1
A ; (13)
where c 2 Q
�
.
P r o o f. We denote
u(t) = W (t; �)
0
@J�1r (a)
tZ
a
W �(s; ��) ~A(s)f(s)ds
1
A :
Let f~z; ~gg 2 L00 �
��E and z(t) = 0 for t � �n. From Lagrange's formula (1),
we obtain
�nZ
a
( ~A(s)g(s); u(s))ds �
�nZ
a
( ~A(s)z(s); f(s))ds = 0:
The equality
�nZ
a
( ~A(s)g(s); y(s))ds �
�nZ
a
( ~A(s)z(s); f(s))ds = 0
is true for every ordered pair f~y; ~fg 2 L�0 � �E. It follows from the last two
equalities that (~g; ~y � ~u)Bn
= 0. Since ker(a; �n; �) is closed and g 2 R(L00 � �E)
is arbitrary, from Lemma 3 and remark we obtain the equality ~y� ~u = ~W (t; �)cn.
Since the interval (a; �n) is taken arbitrarily, we obtain (13) where c = fcng 2 Q
�
.
Note that Lemmas 3, 4 follow also from paper: V.M. Bruk, J. Math. Phys.,
Anal., Geom. 2 (2006), 1�10.
Theorem. Every generalized resolvent R�, Im� 6= 0, of the relation L0 is
the integral operator
R�
~f =
1Z
a
K(t; s; �) ~A(s)f(s)ds ( ~f 2 B):
The kernel K(t; s; �) has the form
K(t; s; �) = ~W (t; �)(M(�) � (1=2)sgn(s� t)J�1r (a))W �(s; ��);
382 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis...
where M(�) : Q+ ! Q
�
is the continuous operator such that M(��) = M�(�) and
(Im�)�1Im(M(�)x; x) � 0 (14)
for every �xed �, Im� 6= 0, and for every x 2 Q+. The operator function M(�)x
is holomorphic for every x 2 Q+ in the semi-planes Im� 6= 0.
P r o o f. First, we prove the theorem for the functions �nite on (a;1) .
Suppose ~f 2 B and f is a �nite function. It follows from (13) that ~y = R�
~f has
the following form:
~y = ~y(t; ~f; �) = ~W (t; �)
0
@c( ~f; �) + (1=2)J�1r (a)
tZ
a
W �(s; ��) ~A(s)f(s)ds
� (1=2)J�1r (a)
1Z
t
W �(s; ��) ~A(s)f(s)ds
1
A ; (15)
where c( ~f; �) 2 Q
�
and c( ~f; �) is uniquely determined by ~f and �, Im� 6= 0.
Indeed, if it is not so, then ~W (t; �)c( ~f ; �) = R�0 = 0, and this equality is true
whenever c( ~f; �) = 0. Therefore, c( ~f; �) = C(�) ~f where C(�) : ~B ! Q
�
is the
linear operator.
Now we show that the operator C(�) is continuous for every �xed �. Let the
sequence f ~fkg ( ~fk 2 Bn) be bounded in Bn for a �xed number n 2 N. Then f ~fkg is
bounded in B. Hence the sequence fR�
~fkg is bounded in B. Consequently, fR�
~fkg
is bounded in Bm. It follows from (15) that the sequence f ~W (t; �)pmc( ~fk; �)g is
bounded in Bm. Since n;m 2 N are arbitrary and according to Lemma 2, it
follows that the operator C(�) is continuous.
Now we prove that c( ~f; �) is uniquely determined by the element W
�
(�) ~f 2
Q+. We assume that W
�
(�) ~f = 0. Then the ordered pair f~z; ~fg 2 L0 � �E,
where
~z(t) = ~W (t; �)
0
@(1=2)J�1r (a)
tZ
a
W �(s; ��) ~A(s)f(s)ds
� (1=2)J�1r (a)
1Z
t
W �(s; ��) ~A(s)f(s)ds
1
A :
Since (L0 � �E)�1 � R�, we obtain that ~W (t; �)c( ~f ; �) belongs to the range of
the operator R�. Hence c( ~f; �) = 0.
Thus C(�) = M(�)W
�
(�) ~f , where M(�) : Q+ ! Q
�
is an everywhere de�ned
operator. We prove that M(�) is continuous for every �xed �. Let the sequence
fqkg = fW
�
(�) ~fkg be bounded in Q+(n). By B
(0)
n we denote the orthogonal
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 383
V.M. Bruk
complement of kerW �
n(�) in the space Bn. The operator W �
n(�) is a continuous
one-to-one mapping of B
(0)
n onto Q+(n). Consequently, there exists a bounded
sequence f~gkg (~gk 2 B
(0)
n ) in Bn such that qk = W
�
(�) ~fk = W �
n(�)~gk. Then the
sequence fR�~gkg is bounded in B. Consequently, fR�~gkg is bounded in Bm for
every m 2 N. Hence the sequence f ~W (t; �)pmM(�)qkg is bounded in Bm. It
follows from Lemma 2 that the operator M(�) is continuous.
Now we prove that the function M(�)x is holomorphic (Im� 6= 0) for every
x 2 Q+. It follows from (15) and holomorphicity of R� that the function � !
~W (t; �)pnC(�) ~f is holomorphic in Bn for every ~f 2 Bj , n; j 2 N. Substitu-
ting pnC(�) ~f for c in equality (4), we get that function � ! ~W (t; 0)pnC(�) ~f
is holomorphic. Since the operator x ! ~W (t; 0)x is a continuous one-to-one
mapping of Q
�
(n) onto ker(a; �n) and ker(a; �n) is closed in Bn, we obtain that
� ! pnC(�) ~f = pnM(�)W �
j
(��) ~f is the holomorphic function for every ~f 2 Bj .
Now holomorphicity of function � ! pnM(�)x follows from the lemma proved
in [11].
Lemma 5. Suppose bounded operators S3(�) : B1 ! B3, S1(�) : B1 ! B2,
S2(�) : B2 ! B3 satisfy the equality S3(�) = S2(�)S1(�) for every �xed � belon-
ging to some neighborhood of a point �0 and suppose the range of operator S1(�0)
coincides with B2, where B1, B2, B3 are Banach spaces. If functions S1(�),
S3(�) are strongly di�erentiable in the point �0, then in this point function S2(�)
is strongly di�erentiable.
In this lemma it should be taken that B1 = Bj , B2 = Q+(j), B3 = Q
�
(n),
S1(�) = W �
j
(��), S2(�) = pnM(�), S3(�) = pnC(�).
So, the operator function � ! pnM(�)x is strongly di�erentiable for every
n 2 N and for every x 2 Q+. Now holomorphicity of the operator function
M(�)x for every x 2 Q+ follows from the closeness of Q
�
in the product of
spaces Q
�
(n) [10, Ch. 2] and from the de�nition of topology of the product
space.
It follows from the equality R�
�
= R�� that M(��) = M�(�) and
~A1=2(s)K�(t; s; �)A1=2(t) = ~A1=2(s)K(s; t; ��)A1=2(t): (16)
Now we show inequality (14). First, we prove the following statement.
Lemma 6. Suppose ~u; ~u0; ~v; ~v0 2 Bn satisfy the equalities
~u(t) = ~W (t; �)(c + J�1r (a)
tR
a
W �(s; ��) ~A(s)u0(s)ds);
~v(t) = ~W (t; �)(d + J�1r (a)
tR
a
W �(s; ��) ~A(s)v0(s)ds);
(17)
384 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
Generalized Resolvents of Symmetric Relations Generated on Semi-Axis...
where d 2 Q
�
(n), c = �J�1r (a)
�nR
a
W �(s; ��) ~A(s)u0(s)ds. Then
�nZ
a
( ~A(t)u0(t); v(t))dt �
�nZ
a
( ~A(t)u(t); v0(t))dt
= �(Jr(a)c; d) � (�� ��)
�nZ
a
( ~A(t)u(t); v(t))dt: (18)
P r o o f. Since Jr(a)c 2 Q+(n), we see that the right-hand side (18) exists.
Let dk 2 Q(n) and the sequence fdkg converges to d as k ! 1 in the space
Q
�
(n). If we replace d by dk in (17), then we obtain the function denoted by
~vk(t). The sequence f~vkg converges to ~v in the space Bn. We apply Lagrange's
formula (1) to the functions u, vk. From the equalities û(�n) = 0, ~A(t)u0(t) =
l[u]�� ~A(t)u, ~A(t)v0(t) = l[vk]�� ~A(t)vk, we obtain the equality of the form (18),
where v is replaced by vk. By calculating to the limit as k !1, we obtain (18).
The lemma is proved.
In order to prove inequality (14), we take the arbitrary element x 2 Q+. Then
there is n 2 N such that x 2 Q+(n). Consequently, there exists ~f 2 Bn such that
�nZ
a
W �(s; ��) ~A(s)f(s)ds = W
�
(�) ~f = x:
Let ~z(t) = ~W (t; �)(M(�)x + (1=2)J�1r (a)x). Suppose ~y = R�
~f has the form
of (15), where c( ~f; �) = M(�)x. Having made some elementary transformations
we can apply Lemma 6 to the functions ~u = ~y � ~z, ~u0 = ~f , ~v = ~y + ~z, ~v0 = ~f .
Then we have
�nR
a
( ~A(t)f; z + y)dt�
�nR
a
( ~A(t)(y � z); f)dt
= 2(x;M(�)x) � (�� ��)
�nR
a
( ~A(t)(y � z); y + z)dt:
Consequently,
(Im�)�1Im(M(�)x; x)
= (z; z)Bn
+ f(�� ��)�1[(R�
~f; ~f)Bn
� ( ~f;R�
~f)Bn
]� (R�
~f;R�
~f)Bn
g: (19)
The operator function R� is a generalized resolvent of the minimal relation
generated in the space Bn by the expression l and the function A(t) (the proof
is similar to the proof of the corresponding statement for the operator from [1]).
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 385
V.M. Bruk
Consequently, the addend in �gurate brackets in the right-hand side (19) is non-
negative. Now (14) follows from (19).
Now we assume that ~f 2 B is not �nite, in general. By V1 we denote the ope-
rator x! ~W (t; �)(M(�)x + (1=2)J�1r (a)x). The operator V1 maps continuously
Q+(n) into B for every n 2 N. Indeed, for any bounded sequence fxkg in Q+(n)
there exists a bounded sequence f~gkg (~gk 2 B
(0)
n ) in Bn such that xk =W �
n(�)~gk.
Then the sequence fR�~gkg is bounded in B. The functions gk vanish out of the
interval [a; �n]. Consequently, the equality
R�~gk = ~W (t; �)(M(�)xk + (1=2)J�1r (a)xk)
is true out of the interval [a; �n]. Therefore the sequence f ~W (t; �)(M(�)xk +
(1=2)J�1r (a)xk)g is bounded in the space B. This implies that the operator V1 is
continuous. Hence we obtain the inequality
1Z
a
( ~A(t) ~W (t; �)(M(�)x + (1=2)J�1r (a)x; ~W (t; �)(M(�)x + (1=2)J�1r (a)x)dt
� k(n; �) kxk2
Q+(n)
; k(n; �) > 0: (20)
Now suppose ~f 2 B and f is a non�nite function, in general. We take the
sequence f ~fng converging to ~f in the space B, where ~fn 2 B and functions fn
are �nite. Using (20) and (16), we obtain that for every �nite function g (~g 2 B)
there exists the limit
lim
n!1
1R
a
( ~A1=2(t)
1R
a
K(t; s; �) ~A(s)fn(s)ds; ~A
1=2(t)g(t))dt
= lim
n!1
1R
a
( ~A1=2(s)fn(s); ~A
1=2(s)
1R
a
K(s; t; ��) ~A(t)g(t)dt)
=
1R
a
( ~A1=2(s)f(s); ~A1=2(s)
1R
a
K(s; t; ��) ~A(t)g(t)dt):
Hence the sequence
�
1R
a
K(t; s; �) ~A(s)fn(s)ds
�
converges to R�
~f as n ! 1
at least weakly in the space B. The theorem is proved.
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