On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. II. Abstract Theory
Special maximal semi-definite subspaces (maximal dissipative and accumulative relations) are considered. Particular cases of those arise in studying boundary problems for systems mentioned in the title. We provide a description of such subspaces and list their properties. A criterion is found that c...
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irk-123456789-1066212016-10-02T03:02:39Z On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. II. Abstract Theory Khrabustovsky, V.I. Special maximal semi-definite subspaces (maximal dissipative and accumulative relations) are considered. Particular cases of those arise in studying boundary problems for systems mentioned in the title. We provide a description of such subspaces and list their properties. A criterion is found that condition of semi-definiteness of sum of indefinite quadratic forms reduces to semi-definiteness of each of the summand forms, i.e it is separated. In the case when the forms depend on a parameter λ (e.g., a spectral parameter) within a domain Λ is in C, a condition is found under which separation of the semi-definiteness property at a single λ implies its separation for all λ. 2006 Article On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. II. Abstract Theory / V.I. Khrabustovsky // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 3. — С. 299-317. — Бібліогр.: 35 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106621 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Special maximal semi-definite subspaces (maximal dissipative and accumulative relations) are considered. Particular cases of those arise in studying boundary problems for systems mentioned in the title. We provide a description of such subspaces and list their properties. A criterion is found that condition of semi-definiteness of sum of indefinite quadratic forms reduces to semi-definiteness of each of the summand forms, i.e it is separated. In the case when the forms depend on a parameter λ (e.g., a spectral parameter) within a domain Λ is in C, a condition is found under which separation of the semi-definiteness property at a single λ implies its separation for all λ. |
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Khrabustovsky, V.I. |
| spellingShingle |
Khrabustovsky, V.I. On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. II. Abstract Theory Журнал математической физики, анализа, геометрии |
| author_facet |
Khrabustovsky, V.I. |
| author_sort |
Khrabustovsky, V.I. |
| title |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. II. Abstract Theory |
| title_short |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. II. Abstract Theory |
| title_full |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. II. Abstract Theory |
| title_fullStr |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. II. Abstract Theory |
| title_full_unstemmed |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. II. Abstract Theory |
| title_sort |
on the characteristic operators and projections and on the solutions of weyl type of dissipative and accumulative operator systems. ii. abstract theory |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106621 |
| citation_txt |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. II. Abstract Theory / V.I. Khrabustovsky // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 3. — С. 299-317. — Бібліогр.: 35 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT khrabustovskyvi onthecharacteristicoperatorsandprojectionsandonthesolutionsofweyltypeofdissipativeandaccumulativeoperatorsystemsiiabstracttheory |
| first_indexed |
2025-07-07T18:47:11Z |
| last_indexed |
2025-07-07T18:47:11Z |
| _version_ |
1837015008559497216 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 3, pp. 299�317
On the Characteristic Operators and Projections
and on the Solutions of Weyl Type of Dissipative
and Accumulative Operator Systems.
II. Abstract Theory
V.I. Khrabustovsky
Ukrainian State Academy of Railway Transport
7 Feyerbakh Sq., Kharkov, 61050, Ukraine
E-mail:khrabustovsky@kart.edu.ua
Received February 3, 2004
Special maximal semi-de�nite subspaces (maximal dissipative and accu-
mulative relations) are considered. Particular cases of those arise in studying
boundary problems for systems mentioned in the title. We provide a descrip-
tion of such subspaces and list their properties. A criterion is found that
condition of semi-de�niteness of sum of inde�nite quadratic forms reduces
to semi-de�niteness of each of the summand forms, i.e it is separated. In the
case when the forms depend on a parameter � (e.g., a spectral parameter)
within a domain � � C , a condition is found under which separation of the
semi-de�niteness property at a single � implies its separation for all �.
Key words: maximal semi-de�nite subspace, maximal dissipative (accu-
mulative) relation, idempotent.
Mathematics Subject Classi�cation 2000: 34B07, 34G10, 46C20, 47A06,
47B50.
This work constitutes Part II of [32]. Notation, de�nitions, numeration of
sections, statements, formulas etc., as well as the list of references, extend those
of [32].
2. A Description and a Properties of Maximal Semi-de�nite
Subspaces of a Special Form
Let Qj = Q�j 2 B(H), Q�1j 2 B(H), j = 1; 2; dimH�(Q1) = dimH�(Q2),
with H�(Qj) being invariant subspaces for the operators Qj , which correspond
c
V.I. Khrabustovsky, 2006
V.I. Khrabustovsky
to positive and negative parts of their spectra. Then it is well know that there
exists �j 2 B(H) such that
��1j 2 B(H); ��jQj�j = J; (2.1)
where J is the canonical symmetry, that is J = J� = J�1 (for example [27], one
can choose �j so that J = sgnQ1 or J = sgnQ2). Represent J in the form
J = P+ � P�; (2.2)
with P� being a pair of complementary orthogonal projections.
Introduce the notation
Q = diag(Q1;�Q2): (2.3)
Let Aj , j = 1; 2, be linear operators inH (possibly unbounded and not densely
de�ned) and suppose DA1
= DA2
= D.
Consider the linear manifold
L = fA1f �A2f jf 2 Dg � H
2 (2.4)
and the operators
S = P+�
�1
1 A1 + P��
�1
2 A2; S1 = P+�
�1
2 A2 � P��
�1
1 A1: (2.5)
Theorem 2.1. L (2.4) is a maximal Q-nonnegative (Q-nonpositive) subspace
in H2 if and only if the following conditions hold:
1o. R(S) = H (R(S1) = H).
2o. There exists a compression K+ (K�) in H such that
S1f = K+Sf (Sf = K�S1f) 8f 2 D: (2.6)
(Under 1o K+ (K�) is unique).
Under (2.6), where linear operators K� are not necessary from B(H), the
operators Aj allow a parametrization as follows:
A1 = �1(P+ � P�K+)S (A1 = �1(P+K� � P�)S1); (2.7)
A2 = �2(P� + P+K+)S (A2 = �2(P+ + P�K�)S1): (2.8)
P r o o f. For certainty, we expound a proof for the case of Q-nonnegative L.
Necessity. Suppose L (2.4) is a maximal Q-nonnegative subspace. Since
U
�J2U = ~J2; (2.9)
300 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On the Characteristic Operators and Projections and on the Solutions of Weyl...
where
U =
�
P+ �P�
P� P+
�
= U
��1; (2.10)
J2 = diag(J;�J); ~J2 = diag(I;�I); (2.11)
the subspace
~L = U
���1L = fSf � S1f j f 2 Dg (2.12)
with
� = diag(�1;�2); (2.13)
is maximal ~J2-nonnegative. If so (see [24, p. 100], [25, Ch. I, � 8]), there exists a
compression K+ in H such that
~L = fg �K+gj g 2 Hg: (2.14)
Compare (2.12), (2.14) to see that 1o and 2o hold.
Su�ciency. Suppose 1o and 2o hold. Multiply from the left both parts of the
initial formulas in (2.5), (2.6) by P+ and P� respectively, and then sum up the
resulting equalities to get the initial equality in (2.7). The initial equality from
(2.8) can be deduced in a similar way.
With the notation
Uj [f ] = (QjAjf;Ajf); f 2 D; (2.15)
apply (2.1), (2.2), (2.7), (2.8) to deduce that
U1[f ]� U2[f ] = kSfk
2
� kK+Sfk � 0; (2.16)
since K+ is a compression. Thus L (2.4) is Q-nonnegative. Prove its maximality.
For that, as one can see from [23], [25, p. 38], in view of (2.1), (2.2), it su�ces to
verify that
P+L = P+H
2; (2.17)
where
P+ = �
�
P+ 0
0 P�
�
��1: (2.18)
Apply (2.18), (2.13), (2.7), (2.8), together with the fact that R(S) = H, to
deduce that
P+L = P+fA1f �A2f jf 2 Dg = �fP+Sf � P�Sf jf 2 Dg
= �fP+g � P�gjg 2 Hg = �fP+g � P�hjg; h 2 Hg = PH
2:
Thus (2.17), along with Th. 2.1, is proved.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 301
V.I. Khrabustovsky
Remark 2.1. Condition 1o in Th. 2.1 in the case dimH =1 could be replaced
in general neither by
9� > 0 : 8f 2 D kSfk � �kfk (kS1fk � �kfk); (2.19)
nor by
9� > 0 : 8f 2 D kA1fk+ kA2fk � �kfk:? (2.20)
P r o o f. Let H = l2. Set up
A1 = �1P+U;A2 = �2P�U (A1 = ��1P�U;A2 = �2P+U)
with U being the one-sided shift in l2 [28]. Then S = U , S1 = 0 (S1 = U , S = 0),
hence condition 2o in Th. 2.1 holds with the compression K+ = 0 (K� = 0).
Therefore, in view of (2.16) (an analog of (2.16) for equality S = K�S1), L (2.4)
is Q-nonnegative (Q-nonpositive). On the other hand, R(S) 6= H (R(S1) 6= H),
although (2.19), (2.20) hold. The Remark 2.1 is proved.
Theorem 2.1 implies
Corollary 2.1. Let the linear manifold L and the operators S, S1 be given by
(2.4), (2.5), and suppose the following two conditions are satis�ed:
1) L is Q-nonnegative (Q-nonpositive).
2) S�1 2 B(H) (S�11 2 B(H)).
Then L is a maximal Q-nonnegative (respectively, Q-nonpositive) subspace.
P r o o f is expounded here, e.g., for the Q-nonnegative case. Verify that
1), 2) imply the Conditions 1o, 2o of Th. 2.1. 2) implies 1o together with (2.6)
in which K+ = S1S
�1. Then with this K+ the representations (2.7), (2.8) are
valid, hence also equality (2.16). On the other hand, 1) implies inequality (2.16),
whence K+ is a compression. The Corollary 2.1 is proved.
Remark 2.2. The transformation
�
iI I
I iI
�
U
���1L
with U, � as in (2.10), (2.13), reduces the maximal Q-nonnegative (Q-nonpositive)
subspace L (2.4) to a maximal accumulative (dissipative) relation in H. Its Cayley
transform V , relates to the compressions K� from Th. 2.1 as follows: V = �iK�.
P r o o f follows from the proof of Th. 2.1 and [22] (see also [2]).
?(2.19))(2.20). If (2.6) holds, where B(H) 3 K� are not necessary compressions, then
(2.20))(2.19).
302 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On the Characteristic Operators and Projections and on the Solutions of Weyl...
Remark 2.3. (cf. [24, 25]). The formulae
L = f�1(P+ � P�K+)h� �2(P� + P+K+)hjh 2 Hg
(L = f�1(P+K� � P�)h� �2(P+ + P�K�)hjh 2 Hg)
(2.21)
establish a one-to-one correspondence between compressions K+ (K�) in H and
maximal Q-nonnegative (Q-nonpositive) subspaces L in H2. (In the case L being
of the form (2.4), the compressions K+ (K�) in (2.7), (2.8) coincide to those in
(2.21)). Besides that:
1) L (2.21) is maximal Q-neutral subspace? if and only if K+ (K�) is an
isometry in H.
2) L (2.21) is hypermaximal Q-neutral subspace if and only if K+ (K�) is a
unitary in H.
P r o o f is expounded here for certainty in the Q-nonnegative case. If L is of
the form (2.21) withK+ being a compression, then this L satis�es the assumptions
of Th. 2.1 since with this L one has S = I, S1 = K+S. Thus by Th. 2.1 L is a
maximal Q-nonnegative subspace.
Conversely, let L be a maximal Q-nonnegative subspace. Then one can use
the idea of the proof of necessity in Th. 2.1 to deduce that L = �U ~L with �, U, ~L
as in (2.13), (2.10), (2.14), and additionally that in (2.14) K+ is a compression,
which implies (2.21).
A classi�cation of L (2.21) in terms of the properties of compressions K�
follows from (2.16) and [24, p. 100], [25, Ch. I, � 4, 8]. Since the correspondence
(2.21) is obviously on-to-one, the statement of the remark is proved.
The following theorem allows one to characterize a maximal Q-de�nite sub-
space in terms of a linear equation, which provides an analog of the existing
characterization for Hermitian [33] (see also [3]) and maximal dissipative or ac-
cumulative [22], (see also [2]) relations.
Theorem 2.2. Suppose that the linear manifold L (e.g. L (2.4)) is a maxi-
mal Q-nonnegative (Q-nonpositive) subspace in H2. Then there exists a unique
compression K+ (K�) in H such that
f � g 2 L , B1f �B2g = 0; (2.22)
where
B1 = (K+P+ � P�)�
�
1Q1; B2 = (K+P� + P+)�
�
2Q2
( B1 = (P+ �K�P�)�
�
1Q1; B2 = (K�P+ + P�)�
�
2Q2 )
(2.23)
?In view of [25, p. 42] maximal Q-neutral subspace is maximal Q-nonnegative or maximal
Q-nonpositive or both type.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 303
V.I. Khrabustovsky
and L admits representation (2.21) with these compressions K�.
If in (2.23) K� are arbitrary compressions in H, then
L̂ = fB�1f �B�2f jf 2 Hg � H
2 (2.24)
is a maximal Q�1-nonpositive (Q�1-nonnegative) subspace in H2 and (as one can
see from (2.23)),
kB�1fk+ kB
�
2fk > 0; 0 6= f 2 H: (2.25)
If L is of the form (2.4) with Aj 2 B(H) and
kA1fk+ kA2fk > 0; 0 6= f 2 H; (2.26)
then S�1 2 B(H), (S�11 2 B(H)), where S, S1 are as in (2.5), hence by (2.6)
one has K+ = S1S
�1 (K� = SS�11 ), i.e. Bj (2.23) admits an explicit expression
in terms of Aj.
Conversely, suppose L is given by (2.22), with Bj 2 B(H), j = 1; 2, and L̂
(2.24) is a maximal Q�1-nonpositive (Q�1-nonnegative) subspace in H2. Then
L is a maximal Q-nonnegative (Q-nonpositive) subspace in H2 (hence admits
representation (2.21)). Furthermore, if (2.25) holds, then the compressions K�
in (2.21) admit explicit expression in terms of Bj, speci�cally K+ = S�1
�1S�
(K� = S��1S�1) with S, S1 being given by (2.5), where A1 = Q�1B�1 , A2 = Q�12 B�2
and S�11 2 B(H) (S�1 2 B(H)).
P r o o f is expounded here for certainty in the Q-nonnegative case. Let L be
a maximal Q-nonnegative subspace. Then by Remark 2.3 there exists a unique
compression K+, which makes valid (2.21), an equivalent of the initial equality in
(2.12) with ~L (2.14). This implies by a virtue of [25, p. 73] that
L
[Q] = Q�1L̂;
with L̂ being as in (2.24), (2.23); L[A] stands here for A-orthogonal complement
in H2. Therefore
f � g 2 L , (Q1f;Q
�1
1 B�1h)� (Q2g;Q
�1
2 B�2h) = 0 8h 2 H;
which implies (2.22), (2.23). Furthermore, Q�1L̂ is of the form (2.21) with K� =
K�
+, hence L̂ (2.24), (2.23) is a maximal Q�1-nonpositive subspace by Remark
2.3.
If L (2.4) with Aj 2 B(H) being a maximal Q-nonnegative subspace, then
R(S) = H by Th. 2.1. Besides that, KerS = f0g since if Sf = 0 for some nonzero
f 2 H, then by condition (2.6) of Th. 2.1 S1f = 0 implies A1f = A2f = 0, which
contradicts (2.26). Thus we have S�1 2 B(H) by the Banach theorem.
304 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On the Characteristic Operators and Projections and on the Solutions of Weyl...
Prove the converse. By our assumption, Q�1L̂ is a maximal Q-nonpositive
subspace. An application of Th. 2.1 provides the existence of a compression K�
such that
Q�11 B�1 = �1(P+K� � P�)S1; Q�12 B�2 = �2(P+ + P�K�)S1;
where S1 is given by (2.5) with Aj being replaced by Q�1j B�j . Note that by a
virtue of 1o of Th. 2.1 one has KerS�1 = f0g, which yields
B1f �B2g = 0 , (K�
�
P+ � P�)�
�
1Q1f � (P+ +K�
�
P�)�2Q2g = 0:
Therefore L = (Q�1L̂)[Q], hence [25, p. 73] L is a maximal Q-nonnegative
subspace. An argument similar to that proving the direct statement demonstrates
that for L in (2.21) operator K+ = K�
�
, which allows to one deduce the rest of
statements in a similar way. The theorem is proved.
Lemma 2.1. (cf. [24, 25]). Let H = H1 �H2; in (2.1) one has
J =
�
I1 0
0 �I2
�
(2.27)
with Ij being the identity operators in Hj, j=1,2. Then the formulae: L = A1H
(L = A2H), where
A1 = �1
�
I1 0
K21 0
�
; A2 = �2
�
0 K12
0 I2
�
; (2.28)
establish a one to one correspondence between compressions K21 2 B(H1;H2)
(K12 2 B(H2;H1)) and maximal Q1-nonnegative (Q2-nonpositive) subspaces L
in H. Besides that:
f 2 L ,
�
0 0
K21 I2
�
��1Q1f = 0
��
I1 K12
0 0
�
��2Q2f = 0
�
:
The Lemma 2.1 proves in the same way as (2.21), (2.22), (2.23) with using
[24, 25].
Note that with H = H1 �H1 and
Q1 = Q2 =
�
0 iI1
�iI1 0
�
;
the maximal Q1-nonnegative (Q1-nonpositive) subspace in H appears to be a
maximal accumulative (dissipative) relation in H1, and, after a suitable change
of notation, Lemma 2.1 provides a well known [22] (see also [2, 3]) description for
them.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 305
V.I. Khrabustovsky
Lemma 2.2. Let H = H1 � H2 and the operator J in (2.1) is just (2.27).
Then L (2.4) together with the operators A1; A2 as in (2.28) of Lemma 2.1 is a
maximal Q-nonnegative subspace in H2.
P r o o f. For L (2.4), (2.28) one has S = I, S1 =
�
0 K12
�K21 0
�
, so Lemma
2.2 is proved in view of Th. 2.1.
An analog for Lemma 2.2 is also valid for the Q-nonpositive case.
In addition to Th. 2.1, we have
Theorem 2.3. Let L (2.4) be a maximal Q-nonnegative (Q-nonpositive) sub-
space in H2 (that is, the assumptions 1o, 2o of Th. 2.1 are satis�ed), and sup-
pose that H = H1 � H2 with the operator J in (2.1) being just (2.27). Then
(�1)j(QjAjf;Ajf) � 0 ((�1)j(QjAjf;Ajf) � 0) for f 2 D, j = 1; 2, if and only
if the compressions in (2.7),(2.8) are of the form
K+ =
�
0 K+
12
K+
21 0
�
;
�
K� =
�
0 K�
12
K�
21 0
��
; (2.29)
with K�
ij 2 B(Hj; being obviously compressions.
P r o o f is to be expounded here for certainty in the Q-nonnegative case.
Necessity. Let (�1)j(QAjf;Ajf) � 0 for f 2 D, j = 1; 2. Then since L (2.4)
is a maximal Q-nonnegative subspace, the linear manifolds fA1f j f 2 Dg and
fA2f j f 2 Dg are, respectively, maximal Q1-nonnegative and Q2-nonpositive
subspaces in H. Thus by Th. 2.1 and Lemma 2.2 one has 8f 2 D 9h 2 H: ?
(P+ � P�K+)Sf =
�
I1 0
K21 0
�
h; (2.30)
(P� + P+K+)Sf =
�
0 K12
0 I2
�
h; (2.31)
where Sf = g1 � g2, h = h1 � h2; gj; hj 2 Hj, and the compression
K+ =
�
K+
11 K+
12
K+
21 K+
22
�
; (2.32)
with K+
ij 2 B(Hj;Hi) .
Multiply (2.30) from left by P+ to get, in view of (2.27),
g1 = h1: (2.33)
?And 8h 2 H9f 2 D :
306 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On the Characteristic Operators and Projections and on the Solutions of Weyl...
In a similar way, multiply (2.30) from left by P� to obtain in view (2.32)
�K+
21g1 �K+
22g2 = K21h1: (2.34)
Since R(S) = H by Th. 2.1, the vectors gj 2 Hj in (2.33), (2.34) are arbitrary.
Thus it follows from (2.33), (2.34) that K+
21 = �K21, K
+
22 = 0. Deduce similarly
from (2.31) that K+
12 = K12, K
+
11 = 0, which proves the necessity.
Su�ciency. Since L (2.4) is a maximal Q-nonnegative subspace in H2, it
follows from Th. 2.1 together with (2.7), (2.8), (2.27), (2.29), that
A1 = �1
�
I1 0
�K+
21 0
�
S; A2 = �2
�
0 K+
12
0 I2
�
S:
So by Lem. 2.1 su�ciency, along with theorem 2.3 is proved.
Consider examples (Th. 2.4�2.7) of Q-semi-de�nite subspaces which arise in
investigation of boundary problems for the equation (0.1).
Let P be an orthogonal projection in H (in particular P can be an orthogonal
projection onto N? (see [32])), and let M�i be a linear operators (not necessary
bounded) in H with the property
M�i = PM�iP (2.35)
(hence also PDM�i
� DM�i
, (I � P )H � DM�i
).
Let G = G� 2 B(H); G�1 2 B(H) (in particular G can be equal to Q(c) (see
[32])).
Represent M�i in the form
M�i =
�
P�i �
1
2
I
�
(iG)�1: (2.36)
Consider linear manifolds in H2:
L�i =
��
(P�i � I)(iG)�1P + (I � P )
�
f �
�
P�i(iG)�1P + (I � P )
�
f jf 2 DM�i
?
(2.37)
and introduce the notation
G2 = diag(G;�G):
?Which are subspaces if and only if the operators M�i are closed.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 307
V.I. Khrabustovsky
Lemma 2.3. If DMi
= H and the operators M�i are related as follows
M�i =M�
i ;
? (2.38)
then the linear manifolds Li and L�i are G2-orthogonal.
P r o o f reduces to a direct computation which uses that, in view of (2.38),
P�i = I �G�1P�i G: (2.39)
Lemma 2.4. The linear manifolds L�i are �G2-nonnegative if and only if
�Im(M�if; f) � 0 for all f 2 DM�i
.
P r o o f reduces to a direct computation.
Theorem 2.4. The linear manifolds L�i (2.37) are maximal �G2-nonnegative
subspaces in H2 if and only if �M�i are maximal dissipative operators in H.
P r o o f is expounded here for certainty in the case of Li. Necessity. Suppose
Li is a maximal G2-nonnegative subspace. Hence operator Mi is closed.
Prove that DMi
= H. Clearly H can be represented in the form H = H1�H2
so that there exists � 2 B(H) with ��1 2 B(H), ��G� = J (2.27). For Li (2.37)
compute the operator S (2.5) with �1 = �2 = �. One has:
�S =Mi +
i
2
���P + I � P: (2.40)
Suppose there exists a nonzero f0 2 D?Mi
. Since R(S) = H by Th. 2.1, there
exists g0 2 DMi
such that �Sg0 = f0. Then it follows from (2.40), (2.35) that
0 = (f0; P g0) = (MiPg0; P g0) +
i
2
k��Pg0k
2;
whence
0 = Im(MiPg0; P g0) +
1
2
k��Pg0k
2: (2.41)
It follows from (2.41) that Pg0 = 0, since the �rst term in (2.41) is nonnegative
by Lemma 2.4. On the other hand, (2.40), (2.35) imply that 0 = (f0; (I�P )g0) =
k(I � P )g0k
2, hence g0 = 0 ) DMi
= H. Thus Mi is closed dissipative operator
(see [34]) by Lemma 2.4.
Prove that Im(M�
i f; f) � 0 for f 2 DM�
i
. Since Li is a maximal G2-
nonnegative subspace, it follows from Lemma 2.3 that L�i (2.37), (2.38) is a
G2-nonpositive linear manifold in view of [25, p. 73]. Thus Lemma 2.4 together
?Alternatively, if DM
�i
= H and Mi =M�
�i
308 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
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with (2.38) implies Im(M�
i f; f) � 0 for f 2 DM�
i
, which proves necessity in view
of [34, p. 109].
Su�ciency. Suppose that Mi (2.35) is maximal dissipative. Hence the linear
manifold Li (2.37) is G2-nonnegative by Lemma 2.4.
Prove that for this manifold the operator S given by (2.5) is such that S�1 2
B(H) where L(2:4) = Li, �j = �.
Prove that 0 6= �p(S) [ �c(S). If not, then there exists a sequence ffng such
that fn 2 D(Mi), kfnk = 1, and �Sfn ! 0, whence in view of (2.40) one has
Im(MiPfn; P fn) +
1
2
k��Pfnk
2
! 0: (2.42)
Since the �rst term in (2.42) is nonnegative due to dissipativity of Mi, it
follows from (2.42) that Pfn ! 0. On the other hand, (2.40), (2.35) imply that
k(I � P )fnk
2 = (�Sfn; (I � P )fn)! 0, hence fn ! 0. The contradiction we get
proves that 0 6= �p(S) [ �c(S).
Prove that 0 =2 �r(S). If not, there exists a nonzero f 2 DM�
i
such that
(�S)�f = 0, since D(�S)� = DM�
i
in view of (2.40). Then by a virtue of (2.40),
(2.35) one has
PM�
i Pf �
i
2
P���f + (I � P )f = 0; (2.43)
whence (I � P )f = 0. Thus by (2.43) one has
Im(M�
i Pf; Pf)�
1
2
k��Pfk = 0: (2.44)
It follows from maximal dissipativity of Mi that the �rst term in (2.44) is
nonpositive [34, p. 109]. Thus by (2.44) Pf = 0, hence f = 0. It follows that
0 =2 �r(S), therefore S
�1 2 B(H), which completes the proof in view of Cor. 2.1.
For P = I;M�i 2 B(H) Th. 2.4 is contained in [1].
Corollary 2.2. If �Mi are maximal dissipative operators in H, then L�i =�
[(P�i � I)G�1f + (I � P )g] � [P�iG
�1f + (I � P )g]jf 2 DM�i
; g 2 H
.
P r o o f follows from the fact that for linear manifolds in the right hand side
the analog of Lemma 2.4 holds.?
Lemma 2.5. Let DMi
= H, the operators M�i be related by (2.38), and the
operators X�ij 2 B(H), j = 1; 2, be related by
X�
�i1Q1Xi1 = G = X�
�i2Q2Xi2: (2.45)
?Note that for these manifolds the analog of Lemma 2.3 also holds.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 309
V.I. Khrabustovsky
Then the linear manifolds
L�i = diag(X�i1;X�i2)L�i (2.46)
are Q-orthogonal, with L�i being as in (2.37).
P r o o f follows from (2.45) and Lemma 2.3.
Lemma 2.6. Suppose ~X�ij ; ~X
�1
�ij 2 B(H), j = 1; 2, and the following three
conditions are satis�ed:
1o. ~L�i are a maximal �G2-nonnegative subspaces in H2.
2o. The subspaces
~L�i = diag( ~X�i1; ~X�i2)~L�i
are �Q-nonnegative.
3o.
� ~X�
�i1Q1
~X�i1 � �G � � ~X�
�i2Q2
~X�i2 (2.47)
Then ~L�i are a maximal �Q-nonnegative subspaces in H2.
P r o o f is presented here for certainty in the case of ~Li. Suppose that ~Li
is not maximal, that is H2 contains a Q-nonnegative subspace T � ~Li. Then
the subspace T1 = diag( ~X�1
i1 ; ~X�1
i2 )T contains ~Li. By a virtue of (2.47) for all
f1 � f2 2 T , one has
(G ~X�1
i1 f1;
~X�1
i1 f1)� (G ~X�1
i2 f1; ~X
�1
i2 f2) � (Q1f1; f1)� (Q2f2; f2) � 0;
since T is Q-nonnegative. Thus T1 is a Q-nonnegative subspace, which contradicts
maximality of ~Li. The lemma is proved.
Theorem 2.5. Suppose Li (L�i) (2.37) is a maximal G2-nonnegative (G2-
nonpositive) subspace in H2, and (2.38) holds. Let for X�ij 2 B(H); j = 1; 2;
(2.45) holds.
Then L�i (Li) (2.46) is Q-nonpositive (Q-nonnegative) manifold in H2.
Additionally, if X�1
ij 2 B(H), X�1
�ij 2 B(H), j = 1; 2, (2.47) for ~X�ij = X�ij
holds with + (-), and the spectrum of either of the operators Yi1, Yi2 does not
cover the unit circle, where
�jY�ij = X�ij; �j 2 B(H); ��1j 2 B(H); ��jQj�j = G; j = 1; 2; (2.48)
(hence in view of (2.45) the spectrum of either of the operators Y�i1, Y�i2 does
not cover the unit circle).
Then L�i (Li) (2.46) is a maximal Q-nonpositive (Q-nonnegative) subspace
in H2.
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P r o o f of Q-semide�niteness for L�i (Li) follows from [25, p. 73] in view of
Lemma 2.5 and Th. 2.4.
The subsequent argument is expounded here for certainty in the case when
condition (2.47) (with +) for ~X+ij = Xij holds. In view of (2.48) we have
Y �i1GYi1 � G � Y �i2GYi2:
Thus by (2.45) one has
Y�i1G
�1Y �
�i1 � G�1 � Y�i2G
�1Y �
�i2;
whence in view of [24, p. 96], we deduce that
Y �
�i1GY�i1 � G � Y �
�i2GY�i2;
since the spectrum of either of the operators Y �
�i1; Y
�
�i2 does not cover the unit
circle.
Hence by (2.48) the condition (2.47) (with �) for ~X�ij = X�ij holds. Finally,
maximality of Li implies maximality for L�i in view of (2.38), Th. 2.4, and [34,
p. 109]. Thus L�i is a maximal Q-nonpositive subspace by Lemma 2.6. The
theorem is proved.
The next theorem allows one to use Remark 1.1 for producing c.o. of a bound-
ary problem for the equation (0.1) with a non-separated boundary condition,
whose special case is the periodic boundary condition.
Theorem 2.6. Suppose:
1o.
�;��1 2 B(H); Q2 = ��Q1�: (2.49)
2o.
U 2 B(H);U�Q1U�Q1 � 0 (� 0): (2.50)
3o. The spectrum of U does not cover the unit circle.
Then L (2.4) with
A1 = I; A2 = ��1U (2.51)
is a maximal Q-nonnegative (Q-nonpositive) subspace in H2.
P r o o f is expounded here for certainty in the Q-nonnegative case. It follows
from (2.49), (2.50) that L (2.4), (2.51) is Q-nonnegative.
Since by (2.1), (2.49)
��2�
�Q1��2 = J; (2.52)
one can set up in (2.1) �1 = ��2
def
= �3. Once this is done, the operator S for L
(2.4), (2.51) acquires the form
S = P+�
�1
3 + P��
�1
3 U: (2.53)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 311
V.I. Khrabustovsky
Prove that S�1 2 B(H). Start with demonstrating that 0 =2 �p(S) [ �c(S). If
not, there exists a sequence ffng such that
fn 2 H; kfnk = 1; Sfn ! 0: (2.54)
It follows from (2.53), (2.54) that
P��
�1
3 fn � ��13 fn ! 0; P+�
�1
3 Ufn � ��13 Ufn ! 0; (2.55)
whence n�
(J��13 Ufn;�
�1
3 Ufn)� (J��13 fn;�
�1
3 fn)
�
�
�
(JP+�
�1
3 Ufn; P+�
�1
3 Ufn)� (JP��
�1
3 fn; P��
�1
3 fn)
� o
! 0: (2.56)
On the other hand, by a virtue of (2.52), the �rst bracket in (2.56) is just
(U�Q1Ufn; fn) � (Q1fn; fn), hence nonpositive in view of (2.50). By (2.2), the
second bracket in (2.56) equals
kP+�
�1
3 Ufnk
2 + kP��
�1
3 fnk
2:
Thus we deduce from (2.56) that
P+�
�1
3 Ufn ! 0; P��
�1
3 fn ! 0;
whence fn ! 0 by (2.55). The contradiction we get proves that 0 6= �p(S)[�c(S).
Prove that 0 =2 �r(S). If not, then for some nonzero f 2 H one has
U
����13 P�f = ����13 P+f: (2.57)
On the other hand, since the spectrum of U does not cover the unit circle, it
follows from [24, p. 96] that
(Q�11 U
����13 P�f;U
����13 P�f) +
�
�(Q�11 ���13 P�f;�
��1
3 P�f)
�
� 0: (2.58)
Now by (2.57), (2.52), (2.2), the �rst term in (2.58) equals kP+fk, while the
second term by (2.52), (2.2) equals kP�fk
2, whence f = 0. Hence 0 2 �r(S),
which �nishes the proof in view of Cor. 2.1.
Remark 2.4. The proof show that condition 3o in the Th. 2.6 is unnecessary,
when Qj � 0 (Qj � 0), j = 1; 2, and when Qj � 0 (Qj � 0), U�1 2 B(H).
If Qj are inde�nite or if Qj � 0 (Qj � 0) it is impossible in general to get rid
of 3o.
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In fact, if Q1 = Q2 = W , U = T , where T , inde�nite W see [24, p.67], then
(2.50), (� 0) holds and hence the linear manifold (2.4), (2.51) is Q-nonnegative,
but for it KerS� 6= f0g. Hence (2.4), (2.51) isn't maximal by Th. 2.1. If H = l2,
Q1 = Q2 = �I(I), U is the one-side shift in l2 [28], then (2.50) (with = 0) holds
and for (2.4), (2.51) KerS�(S�1) 6= f0g. Hence (2.4), (2.51) isn't maximal by
Th. 2.1.
Lemma 2.7. Let Aj, j = 1; 2, be linear operators in H, DAj = D,
(�1)j(QjAjf;Ajf) � 0 (hence L (2.4) is a Q-nonnegative manifold in H2), and
suppose L (2.4) is a maximal Q-nonnegative subspace inH2 (hence Lj = fAjf jf 2
Dg are maximal (�1)jQj-nonpositive subspaces in H). Then
L
[Q] = L
[Q1]
1 �L
[Q2]
2 ; (2.59)
where [A] stands for the A-orthogonal complement in the associated Hilbert sub-
space.
P r o o f. Since L is a maximal Q-nonnegative subspace, one deduces by [25,
p. 73] that L[Q] is a maximal Q-nonpositive subspace:
L
[Q] = (L1 �L2)
[Q]
� L
[Q1]
1 �L
[Q2]
2 ; (2.60)
with L
[Qj]
j being maximal (�1)jQj-nonnegative subspaces by [25, p. 73]. Thus by
an analogue of Lemma 2.2 for the Q-nonpositive case, the subspace in the right
hand side of the inclusion (2.60) is maximal Q-nonpositive. Hence the equality in
(2.59), together with the Lemma, is proved.
The case P = I in Th. 2.4 is supplemented by
Theorem 2.7. Let P be linear operator in H. Set
A1 = P � I; A2 = P: (2.61)
1o. Suppose L (2.4), (2.61) is a maximal G2-nonnegative subspace in H2 ?,
hence, in particular,
(GA1f;A1f)� (GA2f;A2f) � 0; f 2 DP : (2.62)
Let inequality (2.62) is separated, i.e., is equivalent to the pair of inequalities
being simultaneously satis�ed:
(�1)j(GAjf;Ajf) � 0; j = 1; 2; f 2 DP : (2.63)
?By a virtue of Th. 2.4, this is equivalent to maximal dissipativity of Mi (2.36), (2.35),
(Pi = P, P = I), hence DP = H.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3 313
V.I. Khrabustovsky
Then
D
P
2 = DP ; P
2 = P; (2.64)
that is, P is an idempotent.
2o. Conversely, let L (2.4), (2.61) be G2-nonnegative, that is, (2.62) holds,
and let P be an idempotent, i.e., (2.64) holds.
Then (2.62) is separated, that is, (2.63) holds.
P r o o f. 1o. Lemmas 2.7, 2.3 imply
L
[G]
1 �L
[G]
2 = L[G2] �
�
�G�1P�Gg � (I �G�1P�G)gjg 2 DP�G
:
It follows that
L
[G]
1 �
�
G�1P�Ggjg 2 DP�G
;
hence one has
((P � I)f;P�h) = 0; 8f 2 DP ; h 2 DP� : (2.65)
On the other hand, since the operator
M = (P �
1
2
I)(iG)�1 (2.66)
is maximal dissipative by Th. 2.4, P is densely de�ned, closed?, hence [30, p. 335]
P� is densely de�ned, and P�� = P. Thus (2.65) means that (P � I)f 2 DP�� =
DP and
P(P � I)f = 0; 8f 2 DP ;
which proves (2.64).
2o. Set up subsequently in (2.62), (2.61) f = Ph, h 2 DP , and f = (P � I)h,
we obtain (2.63) in view of (2.64). The theorem is proved.
Replace G with �G to see that an analogue for Th. 2.7 is valid for G2-
nonpositive L (2.4), (2.61).
For P 2 B(H) Th. 2.7 is contained in [1].
Remark 2.5. There exists a maximal G2-nonnegative subspace of the form L
(2.4), (2.61), with P being an unbounded idempotent, de�ned densely in H.
In fact, represent M(�) (1.104), (1.103), (1.102) in the form (1.20) and set
P = P(i). As the operator M(�) (1.104) is maximal dissipative if Im� > 0, it
follows from Th. 2.4 that P is the desired idempotent.
Theorem2.7 implies
?Closeness of P also follows from the fact that L (2.4), (2.61) is subspace (see the footnote
to (2.37)).
314 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
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Corollary 2.3. Let for linear operators A1, A2 in H the following conditions
hold: 1) DA1
= DA2
= D; 2) (A2 + A1)
�1 2 B(H) ((A2 � A1)
�1 2 B(H)) and
hence one can de�ne an operator
P = A2(A2 +A1)
�1 (P = A2(A2 �A1)
�1); (2.67)
3) (�1)j(GAjf;Ajf) � 0; f 2 D; j = 1; 2; and hence by Lemma 2.4 an operator M
(2.66), (2.67) is dissipative (see [25]), 4) an operator M (2.66), (2.67) is maximal
dissipative.
Then (2.64) holds for P (2.67).
For A1, A2 2 B(H) Cor. 2.2 is contained in [1].
Next consider L (2.4) with the operators Aj = Aj(�) depending analytically
on �.
Suppose one has operator functions Aj = Aj(�), j = 1; 2, in H (possibly
unbounded and not densely de�ned), with � varying in a domain � � C, and
assume that DAj = D does not depend on j and �.
Lemma 2.8. Suppose that the vector functions Aj(�)f , j = 1; 2, depend an-
alytically on � 2 � for all f 2 D. With S = S(�), S = S1(�) being the vector
functions associated to Aj = Aj(�) by (2.5), assume that for � 2 �:
1o. R(S(�)) = H, (R(S1(�)) = H).
2o. There exists K(�) 2 B(H) such that S1(�) = K(�)S(�) (S(�) =
K(�)S1(�)), with kK(�)k being locally bounded.
Then K(�) depends analytically on � 2 �.
P r o o f is expounded here for certainty in the case S1(�) = K(�)S(�).
First prove that the operator-valued function K(�) is strongly continuous at any
�0 2 �.
Denote by �y an increment of the operator function y = y(�) at �0. For all
f 2 H one has
(�S1)f = (�(KS))f = (�K)S(�0 +��)f +K(�0)(�S)f;
whence
(�K)S(�0 +��)f ! 0 (2.68)
as ��! 0 by continuity of S(�)f and S1(�)f . On the other hand,
k(�K)(�S)fk � k�Kkk(�S)fk ! 0 (2.69)
as �� ! 0 by local boundedness of kK(�)k. It follows from (2.68), (2.69) that
(�K)S(�0)f ! 0 as �� ! 0, hence K(�) is strongly continuous at �0 since
R(S(�0)) = H.
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V.I. Khrabustovsky
Now prove that K(�) is analytic at �0. Since for all f 2 H
�(KS)
��
f =
�K
��
S(�0)f +K(�0 +��)
�S
��
f;
one can take into account that as ��! 0 one has K(�0 +��)f
S
! K(�0),
�(KS)
��
f =
�S1
��
f !
d
d�
(S1f);
�S
��
f !
d
d�
(Sf):
This allows one to deduce that there exists lim
��!0
�K
��
g for all g 2 H since
R(S(�0)) = H. Thus for all g; h 2 H the scalar function (K(�)g; h) is analytic in
the domain �, hence [30, p. 195] K(�) is analytic in �. The Lemma is proved.
Theorem 2.8. Suppose that the vector-functions Ajf = Aj(�)f , j = 1; 2, are
analytic in � 2 �, for all f 2 D, and assume L = L(�) (2.4) for � 2 � is a
maximal Q-nonnegative (Q-nonpositive) subspace, hence, in particular,
U1(�; f)� U2(�; f) � 0 (� 0); � 2 �; (2.70)
with Uj(�; f) = (QjAj(�)f;Aj(�)f), f 2 D.
Then: 1o. If for some � = �0 2 �, for all f 2 D one has an equality in (2.70),
then this equality also holds for all � 2 �.
If, in addition, for some � = �0 2 � and all f 2 D the inequality (2.70) is
separated, i.e., it is equivalent to the following two inequalities being valid simul-
taneously:
U1(�; f) � 0 (� 0); U2(�; f) � 0 (� 0); (2.71)
then (2.70) is separated for all � 2 �.
2o. Suppose that Aj(�) 2 B(H) for � 2 � and (2.26) holds. Then if at some
� = �0 2 � for all nonzero f 2 H one has a strict inequality in (2.70), then the
strict inequality also holds for all � 2 � and all nonzero f 2 H.
P r o o f is expounded here for certainty in the Q-nonnegative case.
1o. By Th. 2.1, Aj = Aj(�) admits representations (2.7), (2.8), with K+ =
K+(�) being a compression in H which depends analytically on � 2 � by Lemma
2.8. If we have an equality in (2.70) at � = �0, then it follows from Remark 2.3
(alternatively, by (2.16)) that K+(�0) is an isometry. Hence one can use e.g., [35,
p. 210] to deduce that K+(�) = K+(�0), for all � 2 �, which implies equality in
(2.70) for all � 2 � by Remark 2.3 (alternatively, by (2.16)).
Suppose that at � = �0 (2.70) is separated. Assume that the operators �j in
(2.7), (2.8) are chosen so that (2.1), (2.27) hold. Then by Th. 2.3, K+(�0) is of
the form (2.29), hence by the above argument, K+(�) = K+(�0) is of the same
316 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 3
On the Characteristic Operators and Projections and on the Solutions of Weyl...
form. Thus by Th. 2.3 the inequality (2.70) is separated for all � 2 �, which
proves 1o.
20. Suppose that for � = �0, for all nonzero f 2 H one has strict inequality in
(2.70), but there exist � =
0 2 � and a nonzero f = f0 2 H which make (2.70) an
equality. Thus kK+(
0)S(
0)f0k = kS(
0)f0k by (2.16), where S�1(�) 2 B(H)
for all � 2 � in view of Th. 2.2. Hence it follows from [35, p. 210] that for all
� 2 �
K+(�)S(
0)f0 = S(
0)f0;
whence
K+(�0)S(�0)g0 = S(�0)g0: (2.72)
with g0 = S�1(�0)S(
0)f0 6= 0. Now (2.72) implies that (2.70) becomes equality
with � = �0, f = g0 in view of (2.16) . The contradiction we get demonstrates
that 2o and the theorem are proved.
Remark 2.6. Suppose we are under assumptions of Th. 2.8 which precede its
no1o, and suppose that for all � 2 � (2.70) (� 0 or � 0) is a strict inequality
with some f = f(�) 2 D. Then the assumption that (2.70) is separated for some
� = �0 2 � does not imply its separation for all � 2 �.
In fact, let
Q1 = Q2 =
�
0 i
�i 0
� �
Q1 = Q2 =
�
0 �i
i 0
��
;
A1 =
�
�� i �+ i
0 1
�
; A2 =
�
��� i i� �
1 0
�
: (2.73)
Then with Im� > 0, L (2.4), (2.73) is a maximal Q-positive (Q-negative)
subspace such that the associated inequality (2.70) separates only at � = i.
References
[32] V.I. Khrabustovsky, On the Characteristic Operators and Projections and on the So-
lutions of Weyl Type of Dissipative and Accumulative Operator Systems. I. General
Case. � J. Math. Phys. Anal. Geom. 2 (2006), No. 2, 149�175.
[33] F.S. Rofe-Beketov, Self-adjoint Extensions of Di�erential Operators in a Space of
Vector-Valued Functions. � Teor. Funkts. Funkts. Anal. i Prilozh. (1969), No. 8,
3�24. (Russian)
[34] S.G. Krein, Linear Di�erential Equations in Banach Space. Nauka, Moscow, 1967.
(Russian). (Engl. Transl.: Math. Monogr. by J.M. Daskin; AMS Transl. 29 (1971),
Providence, RI, v+390.)
[35] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators in Hilbert Space. Mir,
Moscow, 1970. (Russian)
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