On Spaces of Boundary Values for Relations Generated by a Formally Selfadjoint Expression and a Nonnegative Operator Function
The space of boundary values for a maximal relation generated by a formally selfadjoint di erential expression and a nonnegative operator function is constructed.
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irk-123456789-1066182016-10-02T03:02:26Z On Spaces of Boundary Values for Relations Generated by a Formally Selfadjoint Expression and a Nonnegative Operator Function Bruk, V.M. The space of boundary values for a maximal relation generated by a formally selfadjoint di erential expression and a nonnegative operator function is constructed. 2006 Article On Spaces of Boundary Values for Relations Generated by a Formally Selfadjoint Expression and a Nonnegative Operator Function / V.M. Bruk // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 3. — С. 268-277. — Бібліогр.: 11 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106618 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The space of boundary values for a maximal relation generated by a formally selfadjoint di erential expression and a nonnegative operator function is constructed. |
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Article |
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Bruk, V.M. |
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Bruk, V.M. On Spaces of Boundary Values for Relations Generated by a Formally Selfadjoint Expression and a Nonnegative Operator Function Журнал математической физики, анализа, геометрии |
| author_facet |
Bruk, V.M. |
| author_sort |
Bruk, V.M. |
| title |
On Spaces of Boundary Values for Relations Generated by a Formally Selfadjoint Expression and a Nonnegative Operator Function |
| title_short |
On Spaces of Boundary Values for Relations Generated by a Formally Selfadjoint Expression and a Nonnegative Operator Function |
| title_full |
On Spaces of Boundary Values for Relations Generated by a Formally Selfadjoint Expression and a Nonnegative Operator Function |
| title_fullStr |
On Spaces of Boundary Values for Relations Generated by a Formally Selfadjoint Expression and a Nonnegative Operator Function |
| title_full_unstemmed |
On Spaces of Boundary Values for Relations Generated by a Formally Selfadjoint Expression and a Nonnegative Operator Function |
| title_sort |
on spaces of boundary values for relations generated by a formally selfadjoint expression and a nonnegative operator function |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/106618 |
| citation_txt |
On Spaces of Boundary Values for Relations Generated by a Formally Selfadjoint Expression and a Nonnegative Operator Function / V.M. Bruk // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 3. — С. 268-277. — Бібліогр.: 11 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT brukvm onspacesofboundaryvaluesforrelationsgeneratedbyaformallyselfadjointexpressionandanonnegativeoperatorfunction |
| first_indexed |
2025-07-07T18:46:55Z |
| last_indexed |
2025-07-07T18:46:55Z |
| _version_ |
1837014991035695104 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 3, pp. 268�277
On Spaces of Boundary Values for Relations
Generated by a Formally Selfadjoint Expression
and a Nonnegative Operator Function
V.M. Bruk
Saratov State Technical University
77 Politechnitseskaja, Saratov, 410054, Russia
E-mail:bruk@san.ru.
Received May 10, 2005
The space of boundary values for a maximal relation generated by a for-
mally selfadjoint di�erential expression and a nonnegative operator function
is constructed.
Key words: symmetric relation, space of boundary values, formally self-
adjoint di�erential expression.
Mathematics Subject Classi�cation 2000: 47A06, 34B05.
1. Introduction
In a well-known paper by F.S. Rofe-Beketov [1], selfadjoint extensions of a mi-
nimal operator generated by a formally selfadjoint di�erential expression are de-
scribed in the terms of boundary values. The paper stimulated appearance of
numerous works generalizing the obtained results [1] in various directions. The
papers [2�4] are among them. In [2] dissipative extensions of minimal operators
are described, and in [3, 4] abstract spaces of boundary values are introduced.
We construct here a space of boundary values in the context of [3, 4] for
a linear relation generated by a formally selfadjoint di�erential expression and
a nonnegative operator function. These relations were de�ned and explored in [5]
for the �nite-dimensional case and in [6] for the in�nite-dimensional case. The
situation is more complicated in the in�nite-dimensional case. The domain of
maximal relation contains functions the values of which do not belong to the
source space. So, there are some ordered pairs belonging to the maximal relation
such that the Lagrange formula is not valid. In [6] it is assumed that the domain
does not contain these functions. We do not assume this condition here.
c
V.M. Bruk, 2006
On Spaces of Boundary Values for Linear Relations
2. Notations and Auxiliary Statements
Let H be a separable Hilbert space with the scalar product (�; �) and the
norm k�k, then A(t) be an operator function strongly measurable on the compact
interval [a; b] ; the values of A(t) be 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 to be integrable on the interval [a; b].
By l we denote the di�erential expression of order r (r = 2n or r = 2n+ 1):
l[y] =
8>>>>>>><
>>>>>>>:
nX
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;
nX
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 r = 2n and q0(t) in the case r = 2n+1; have the bounded inverse
operator almost everywhere. The functions pn�k(t) are strongly di�erentiable k
times and the functions qn�k(t) are strongly di�erentiable k times in the case
r = 2n, as well as 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 stated.
According to [7] we treat l as a quasidi�erential expression. The quasiderivatives
for the expression l are de�ned in [7]. Suppose that 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 case r = 2n);
q00(t); q1(t); : : : ; qn(t); p0(t); : : : ; pn(t) (in case r = 2n+ 1)
are integrable on the interval [a; b]. Thus we consider the case when the expression
l is regular on the interval [a; b].
We de�ne the scalar product
hy; zi =
bZ
a
(A(t)y(t); z(t))dt;
where y(t), z(t) are H-valued functions that are continuous on [a; b]. We identify
with zero the functions y such that hy; yi = 0 and, having made the completion,
obtain Hilbert space denoted by B = L2(H;A(t); a; b). Let ~y be some element
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 269
V.M. Bruk
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 denote the class of functions containing y. Suppose y 2 ~y. Without loss of
generality, we will say often that y(t) belongs to B.
Let G(t) be the set of elements x 2 H such that A(t)x = 0, 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 that H� (t) (�1 < � < 1) is the Hilbert scale of
spaces generated by the operator A�10 (t). For �xed t; the operator A
1=2
0 (t) is the
continuous one-to-one mapping of H(t) = H0(t) onto H1=2(t). By Â
1=2
0 (t) denote
the operator adjoint to A
1=2
0 (t). The operator Â
1=2
0 (t) is the 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 the continuous one-to-one mapping
of H
�1=2(t) onto H1=2(t) and ~A0(t) is an extension of A0(t). By ~A(t) denote the
operator de�ned on H
�1=2(t)�G(t) such that ~A(t) is equal to ~A0(t) on H
�1=2(t)
and ~A(t) is equal to zero on G(t) . The operator ~A(t) is an extension of A(t).
In [6] it is proved that spaces H
�1=2(t) are measurable with respect to the pa-
rameter t [8, Ch. 1] whenever we take the functions of the form ~A�10 (t)A1=2(t)h(t)
in place of the measurable functions, where h(t) is a measurable function with
the values in H. The space B is the measurable sum of the spaces H
�1=2(t).The
space B consists of elements (i.e., classes of functions) with the representatives of
the form ~A�10 (t)A1=2(t)h(t) , where h(t) 2 L2(H; a; b), i.e.,
R
b
a
kh(t)k2 dt <1.
We de�ne the minimal and maximal relations generated by the expression
l and the function A(t) in the following way. Let D0 be the set of functions
y satisfying the conditions: a) the quasiderivatives 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 we assign the class of functions identi�ed
in B with ~A�10 (t)l[y]. This correspondence may be not an operator because it
can be occurred that some function y is identi�ed with zero in B and ~A�10 (t)l[y]
isn't equal to zero. So, we get the linear relation L0 in the space B. The closure
of L0 we denote by L: The relation L is called maximal. By L0 denote the
restriction of L to the set of elements ~y 2 B with representatives y 2 D0 such that
y[k](a) = y[k](b) = 0 (k = 0; 1; : : : ; r � 1). The relation L0 is called minimal.
Terminology concerning linear relations can be found in the monographs [9�
11]. In what further we will use the following notations: R is the range of values;
f�; �g is the ordered pair; kerL is the set of elements z such that fz; 0g 2 L.
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
270 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On Spaces of Boundary Values for Linear Relations
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 the 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 unmarked elements are equal to zero. (In the matrix J2n+1(t) the
element 2iq�10 (t) stands on the intersection of the row n + 1 and the column
n+1.) Suppose y; z 2 D0, then the Lagrange formula takes in these notations the
following form:
�Z
�
(l[y]; z)dt �
�Z
�
(y; l[z])dt = (Jr(t)ŷ(t); ẑ(t))j
�
�; a 6 � < � 6 b: (1)
It follows from �the method of the variation of arbitrary constants� that the
general solution of the equation l[y]�� ~A(t) = ~A(t)f(t) is represented in the form:
y(t; f; �) = W (t; �)
0
@x+
tZ
a
V (s; �) ~A(s)f(s)ds
1
A ; (2)
where x 2 Hr, V (t; �) is the operator of H to Hr for �xed t, �, and V (t; �)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 271
V.M. Bruk
satis�es the condition
Ŵ (t; �)V (t; �) =
0
BB@
0
� � �
0
�E
1
CCA : (3)
It follows from (2), (3) that
ŷ(t; f; �) = Ŵ (t; �)
0
@x+
tZ
a
V (s; �) ~A(s)f(s)ds
1
A : (4)
Using (1), we obtain
Ŵ �(t; ��)Jr(t)Ŵ (t; �) = Jr(a): (5)
It follows from (3), (5) that
V (t; �) = J�1r (a)W �(t; ��): (6)
Let Q0 be the set of elements x 2 Hr such that the function W (t; 0) is identi-
�ed with zero in the space B, i.e.,
R
b
a
A1=2(s)W (s; 0)x
2 ds = 0. It follows from
the equalities
W (t; �)x = W (t; 0)
0
@x+ �
tZ
a
V (s; 0) ~A(s)W (s; �)xds
1
A ; (7)
W (t; 0)x = W (t; �)
0
@x� �
tZ
a
V (s; �) ~A(s)W (s; 0)xds
1
A (8)
that the function W (t; �)x is identi�ed with zero in the space B if and only if
x 2 Q0 (in the �nite-dimensional case this fact was obtained in [7]).
Let Q be the orthogonal complement of Q0 in Hr, Hr = Q �Q0. We de�ne
the norm
kxk
�
=
0
@
bZ
a
A1=2(s)W (s; 0)x
2
1
A
1=2
6 c kxk ; x 2 Q; (9)
in the space Q. By Q
�
denote the completion of Q with respect to this norm. It
follows from equalities (7), (8) that we obtain the same set Q
�
with the equivalent
norm whenever we replace W (s; 0) by W (s; �) in (9).
272 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On Spaces of Boundary Values for Linear Relations
Suppose the sequence fxng (xn 2 Q) converges to x0 2 Q
�
in Q and sup-
pose ~W (t; �)xn is a class of functions with the representative W (t; �)xn. Then
f ~W (t; �)xng is the fundamental sequence in B. Hence f ~W (t; �)xng converges
to some element belonging to B. We denote this element by ~W (t; �)x0. If the
sequence f ~W (t; �)xng (xn 2 Q) converges in B, then there exists a unique ele-
ment x0 2 Q
�
such that ~y = ~W (t; �)x0. Indeed, it follows from (9) that the
sequence fxng is fundamental in Q
�
, and we can take lim
n!1
xn in place of x0.
Hence ~W (t; �)x0 belongs to ker(L� �E) for every x0 2 Q
�
and ~W (t; �)x0 6= 0 in
B for x0 2 Q
�
, x0 6= 0.
The space Q
�
can be treated as a negative one with respect to Q. By Q+
we denote the corresponding space with the positive norm (see [9, Ch. 2]). Let
W0(�) denote the operator x ! ~W (t; �)x and let W1(�) denote the operator
~f !
R
b
a
W �(s; ��) ~A(s)f(s)ds. Then
W �
0 (�) = W1(��): (10)
The operator W0(�) is the continuous one-to-one mapping of Q
�
into B
for every �xed � and the range of W0(�) is closed. This implies that W1(�) maps
continuously B onto Q+. So, R(W1(�)) = Q+ for every �.
Lemma 1. The relation L consists of all ordered pairs f~y; ~fg 2 B � B such
that
~y = ~W (t; 0)x+ ~F ; (11)
where x 2 Q
�
, ~F is the class of functions identi�ed in B with the function
F (t) = W (t; 0)
tZ
a
V (s; 0) ~A(s)f(s)ds: (12)
P r o o f. Assume that the ordered pair f~y; ~fg 2 B � B satis�es (11), (12).
It follows from the reasoning before Lemma 1 that f~y; ~fg 2 L. Also we note that
f~y; ~fg 2 L0 whenever x 2 Q in (11).
Now we assume that the ordered pair f~y; ~fg 2 L. Then there exists a sequence
of the ordered pair f~yn; ~fng 2 L0 such that f~yn; ~fng converges to f~y; ~fg in B� B
as n!1 and the function yn is represented in the form
yn(t) = W (t; 0)
0
@xn +
tZ
a
V (s; 0) ~A(s)fn(s)ds
1
A ; (13)
where xn 2 Q. Since the sequence f~yn; ~fng converges in B � B, we see that the
sequence f ~W (t; 0)xng converges in B. By taking the limit n ! 1 in (13), we
obtain (11). The lemma is proved.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 273
V.M. Bruk
R e m a r k 1. Since
V (s; 0)A1=2(s)
2 L2(a; b),
A1=2(s)f(s)
2 L2(a; b),
we see that the integral in (12) exists and this integral is continuous with respect
to t.
R e m a r k 2. For any ordered pair f~y; ~fg 2 L there exists a unique element
x 2 Q
�
and a unique function F (t) of form (12) such that equality (11) is true.
Indeed, if f1, f2 are identi�ed in B, then ~A(t)f1 = ~A(t)f2. To conclude the proof,
it remains to note that ~W (t; 0)x 6= 0 in B for x 2 Q
�
and x 6= 0.
R e m a r k 3. The relation L � �E consists of ordered pairs f~y; ~fg of the
form (11), (12), where W (t; 0) and V (t; 0) are replaced by W (t; �) and V (t; �)
respectively. Hence the operator x ! ~W (t; �)x is the continuous one-to-one
mapping of Q
�
onto ker(L� �E).
Lemma 2. L0 is the closed symmetric relation.
P r o o f. Suppose the sequence of ordered pairs f~yn; ~fng 2 L0 converges to
f~y; ~fg in B�B. It follows from the de�nition of L0 that we can take representatives
yn, fn of classes of functions ~yn, ~fn such that (13) is true and ŷn(a) = ŷn(b) = 0.
Using (4), we get in (13) xn =
R
b
a
V (s; 0) ~A(s)fn(s)ds = 0. Calculating the limit
n!1 in this equality and in (13), we obtain that x = 0 in (11), (12) and
bZ
a
V (s; 0) ~A(s)f(s)ds = 0: (14)
It follows from (4) and (14)that y 2 D0 and ŷ(a) = ŷ(b) = 0. This implies that
f~y; ~fg 2 L0. Using the Lagrange formula (1), we get that L0 is the symmetric
relation. The lemma is proved.
R e m a r k 4. It follows from the proof Lemma 2 that the range R(L0) of the
relation L0 consists of all elements ~f 2 B such that (14) true. Therefore R(L0)
is closed and L�10 is the bounded operator on R(L0). Substituting V (s; �) for
V (s; 0), we obtain the analogous statements for the relation L0 � �E. Then (6)
implies that the range R(L0��E) of the relation L0��E consists of all elements
~f 2 B such that
bZ
a
W �(s; ��) ~A(s)f(s)ds = 0:
Thus, R(L0 � �E) = kerW1(�).
By N� we denote the defect subspace of the relation L0, i.e., N� is the ortho-
gonal complement of the range R(L0��E) of the relation L0��E in the space B.
Lemma 3. N� = ker(L� ��E).
274 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On Spaces of Boundary Values for Linear Relations
P r o o f. It follows from Remarks 3, 4 and equality (10) that orthogonal
complement of kerW1(�) = R(L0��E) in B coincides with R(W0(��)) = ker(L�
��E). This completes the proof of Lemma 3.
Lemma 4. L�0 = L.
P r o o f. Using the Lagrange formula (1), we get L0 � L�0. Consequently,
L � L�0. It is known (see [11]) that L�0 is the direct sum of the subspaces L0, ~N�,
~N��: L
�
0 = L0 _+ ~N�
_+ ~N��, where
~N� is the set of ordered pairs of the form fz; ��zg,
z 2 N�. Since L0 � L, ~N� � L, ~N�� � L, we obtain L�0 � L. The lemma is
proved.
3. The Main Result
In what further, we denote W (t; 0) = W (t) for the simpli�cation of notations.
Let f~y; ~fg 2 L. It follows from Lemma 1 and formulae (6), (12) that
~y = ~W (t)c+ ~F , where c 2 Q
�
,
F (t) = W (t)J�1r (a)
tZ
a
W �(s) ~A(s)f(s)ds:
We denote
W1(0) ~f =
Z
b
a
W �(s) ~A(s)f(s)ds = d 2 Q+
and de�ne a pair of boundary values fY; Y 0g 2 Q
�
�Q+ for the pair f~y; ~fg 2 L
by the formulae
Y = �1f~y; ~fg = c+ (1=2)J�1r (a)d 2 Q
�
; Y 0 = �2f~y; ~fg = d 2 Q+: (15)
It follows from Remark 2 that a pair of boundary values fY; Y 0g is uniquely
determined by the pair f~y; ~fg 2 L. We denote by � the operator taking f~y; ~fg 2 L
to fY; Y 0g 2 Q
�
�Q+ by formulae (15).
Theorem. The range R(�) of the operator � coincides with Q
�
� Q+ and
"the Green formula" is valid:
h ~f1; ~y2i � h~y1; ~f2i = (Y 01 ; Y2)� (Y1; Y
0
2); (16)
where f~y1; ~f1g; f~y2; ~f2g 2 L, �f~y1; ~f1g = fY1; Y
0
1g, �f~y2;
~f2g = fY2; Y
0
2g.
P r o o f. Using Lemma 1 and the equality R(W1(0)) = Q+, we obtain the
�rst part of the theorem. Now we shall prove (16). It follows from Lemma 1 that
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 275
V.M. Bruk
~yi = ~zi + ~Fi, where ~zi = ~W (t)ci (ci 2 Q
�
), ~Fi is the class of functions with the
representative
Fi(t) = W (t)J�1r (a)
tZ
a
W �(s) ~A(s)fi(s)ds; i = 1; 2:
We denote di = �2f~yi; ~fig. Then ci = �1f~yi; ~fig� (1=2)J�1r (a)di. The expres-
sion l is de�ned on the functions Fi(t) and l[Fi] = ~A(t)fi. Using the Lagrange
formula (1) and equalities (4), (5), we get
h ~f1; ~F2i � h ~F1; ~f2i = (Jr(b)Ŵ (b)J�1r (a)d1; Ŵ (b)J�1r (a)d2)
= (Ŵ �(b)Jr(b)Ŵ (b)J�1r (a)d1; J
�1
r (a)d2)
= (Jr(a)J
�1
r (a)d1; J
�1
r (a)d2) = (d1; J
�1
r (a)d2): (17)
We take two sequences fci;ng (ci;n 2 Q; i = 1; 2) such that ci;n converges to ci
in Q
�
as n!1. We denote zi;n = W (t)ci;n. The sequence f~zi;ng converges to ~zi,
i = 1; 2, in B. The expression l is de�ned on the functions zi;n(t) and l[zi;n] = 0.
It follows from the Lagrange formula (1) and equalities (4), (5) that
h ~f1; ~z2:ni = h ~f1; ~z2ni � h ~F1; 0i = (Jr(b)Ŵ (b)J�1r (a)d1; Ŵ (b)c2;n)
= (Ŵ �(b)Jr(b)Ŵ (b)J�1r (a)d1; c2;n) = (Jr(a)J
�1
r (a)d1; c2;n) = (d1; c2;n):
We calculate the limit n!1 in this equality. Using d1 2 Q+, we obtain
h ~f1; ~z2i = (d1; c2); (18)
and analogously
h~z1; ~f2i = (c1; d2): (19)
It follows from the equality (J�1r (a)d1; d2) = �(d1; J
�1
r (a)d2) and (17)�(19)
that
h ~f1; ~y2i � h~y1; ~f2i = h ~f1; ~z2i+ h ~f1; ~F2i � h~z1; ~f2i � h ~F1; ~f2i
= (d1; J
�1
r (a)d2) + (d1; c2)� (c1; d2)
= (d1; c2 + (1=2)J�1r (a)d2)� (c1 + (1=2)J�1r (a)d1; d2) = (Y 01 ; Y2)� (Y1; Y
0
2):
The theorem is proved.
Thus it follows from the theorem that the ordered four (Q
�
; Q+;�1;�2) are
the space of boundary values for the relation L in the sense of [3, 4, 6]. In [3,
4] an abstract space of boundary values is introduced for operators and in [6] it
is introduced for relations. The case Q+ = Q
�
is considered in these papers.
276 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On Spaces of Boundary Values for Linear Relations
The detailed bibliography related to the topics of our paper is represented in the
monographs [9, 10].
References
[1] F.S. Rofe-Beketov, Selfadjoint Extensions of Di�erential Operators in a Space of
Vector Functions. � Dokl. Akad. Nauk USSR 184 (1969), No. 5, 1034�1037. (Rus-
sian)
[2] M.L. Gorbatchuk, A.N. Kochubei, and M.A. Ribak, Dissipative Extensions of Dif-
ferential Operators in a Space of Vector Functions. � Dokl. Akad. Nauk USSR 205
(1972), No. 5, 1029�1032. (Russian)
[3] A.N. Kochubei, On Extensions of Symmetric Operators and Symmetric Binary
Relations. � Mat. Zametki 17 (1975), No. 1, 41�48. (Russian)
[4] V.M. Bruk, On One Class of Boundary Value Problems with a Spectral Parameter
in the Boundary Condition. � Mat. Sb. 100 (1976), No. 2, 210�215. (Russian)
[5] V.M. Bruk, On a Number Linearly Independent Square-Integrable Solutions of
Systems of Di�erential Equations. � Funct. Anal., Ulyanovsk 5 (1975), 25�33.
(Russian)
[6] V.M. Bruk, On the Linear Relations in the Space of Vector Functions. � Mat.
Zametki 24 (1978), No. 4, 499�511. (Russian)
[7] V.I. Kogan and F.S. Rofe-Beketov, On Square-Integrable Solutions of Symmet-
ric Systems of Di�erential Equations of Arbitrary Order. � In: Proc. Roy. Soc.
Edinburgh. A 74 (1975), 5�40.
[8] J.L. Lions and E. Magenes, Problemes aux Limities non Homogenenes et Applica-
tions. Dunod, Paris, 1968.
[9] V.I. Gorbatchuk and M.L. Gorbatchuk, Boundary Value Problems for Di�erential-
Operator Equations. Dordrecht�Boston�London, Kluwer Acad. Publ. (1991).
[10] F.S. Rofe-Beketov and A.M. Kholkin, Spectral Analysis of Di�erential Operators.
World Sci. Monogr., Ser. Math., Vol. 7.
[11] E.A. Coddington, Extension Theory of Formally Normal and Symmetric Subspaces.
� Mem. Am. Math. Soc. 134 (1973), 1�80.
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