On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. I. General Case
In the context of dissipative and accumulative di erential equations (which contain the spectral parameter λ nonlinearly) in a separable Hilbert space H we introduce a characteristic operator M(λ) that works as an analog of the characteristic Weyl-Titchmarsh matrix. Its existence and properties are...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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irk-123456789-1065892016-10-01T03:02:14Z On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. I. General Case Khrabustovsky, V.I. In the context of dissipative and accumulative di erential equations (which contain the spectral parameter λ nonlinearly) in a separable Hilbert space H we introduce a characteristic operator M(λ) that works as an analog of the characteristic Weyl-Titchmarsh matrix. Its existence and properties are investigated. A description of M(λ) that corresponds to separated boundary conditions is given. Analogs for Weyl functions and solutions are introduced. Weyl type inequalities for those analogs are established, which reduce to well-known inequalities in various special cases. The proofs are based on description and properties of maximal semi-definite subspaces in H² of special form that we provide while studying boundary problems for equations as above. 2006 Article On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. I. General Case / V.I. Khrabustovsky // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 149-175. — Бібліогр.: 31 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106589 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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In the context of dissipative and accumulative di erential equations (which contain the spectral parameter λ nonlinearly) in a separable Hilbert space H we introduce a characteristic operator M(λ) that works as an analog of the characteristic Weyl-Titchmarsh matrix. Its existence and properties are investigated. A description of M(λ) that corresponds to separated boundary conditions is given. Analogs for Weyl functions and solutions are introduced. Weyl type inequalities for those analogs are established, which reduce to well-known inequalities in various special cases. The proofs are based on description and properties of maximal semi-definite subspaces in H² of special form that we provide while studying boundary problems for equations as above. |
| format |
Article |
| author |
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. I. General Case 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. I. General Case |
| title_short |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. I. General Case |
| title_full |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. I. General Case |
| title_fullStr |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. I. General Case |
| title_full_unstemmed |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. I. General Case |
| title_sort |
on the characteristic operators and projections and on the solutions of weyl type of dissipative and accumulative operator systems. i. general case |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106589 |
| citation_txt |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems. I. General Case / V.I. Khrabustovsky // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 149-175. — Бібліогр.: 31 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT khrabustovskyvi onthecharacteristicoperatorsandprojectionsandonthesolutionsofweyltypeofdissipativeandaccumulativeoperatorsystemsigeneralcase |
| first_indexed |
2025-07-07T18:44:18Z |
| last_indexed |
2025-07-07T18:44:18Z |
| _version_ |
1837014830568964096 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 2, pp. 149�175
On the Characteristic Operators and Projections
and on the Solutions of Weyl Type of Dissipative
and Accumulative Operator Systems. I. General Case
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
In the context of dissipative and accumulative di�erential equations (which
contain the spectral parameter � nonlinearly) in a separable Hilbert space
H we introduce a characteristic operator M(�) that works as an analog
of the characteristic Weyl�Titchmarsh matrix. Its existence and proper-
ties are investigated. A description of M(�) that corresponds to separated
boundary conditions is given. Analogs for Weyl functions and solutions are
introduced. Weyl type inequalities for those analogs are established, which
reduce to well-known inequalities in various special cases. The proofs are
based on description and properties of maximal semi-de�nite subspaces in
H2 of special form that we provide while studying boundary problems for
equations as above.
Key words: operator di�erential equation, characteristic operator, cha-
racteristic projection, solution of Weyl type, maximal semi-de�nite subspace.
Mathematics Subject Classi�cation 2000: 34A55, 34G10, 47E05.
This work shares the subjects of [1] and considers the operator di�erential
equation in a separable Hilbert space H as follows:
i
2
�
(Q(t)x(t))0 +Q(t)x0(t)
��H�(t)x(t) = w�(t)f(t); t 2 �I;I = (a; b) � R1;
(0.1)
with Q(t) = Q�(t), Q�1(t), H�(t) 2 B(H); Q(t) 2 ACloc; the operator-valued
function H�(t) = H�
��
(t) being Nevanlinna; the weight w�(t) = ImH�(t)=Im� � 0
(Im� 6= 0) (see a more detailed description of the properties of H�(t), w�(t) in
Sect. 1, the notions of ACloc are treated in the sense of [2, 3]). Since the weight
c
V.I. Khrabustovsky, 2006
V.I. Khrabustovsky
w�(t) can be degenerate, these considerations cover many types of dissipative and
accumulative di�erential and di�erence equations, as it is demonstrated in [4, 5].
In Section 1 a characteristic operator (c.o.) for the equation (0.1) is intro-
duced, its existence (Th. 1.2) and properties (Th. 1.1) are established. In this
setting, the results of [4, 6�11] are covered (a more detailed exposition of those
works is presented in [1]). In Section 3 we present necessary and su�cient con-
ditions (Th. 3.1) for c.o. correspond to separated boundary conditions, which
reduce with constant Q(t) to conditions established in [1].* Those conditions are
in claiming that c.o. admits a special expression via a projection, which is called
characteristic. Under constant Q(t), a part of these results has been announced
in [1] without proofs.
In Section 4 we introduce operator analogs for Weyl functions and solutions
of (0.1); those are used to describe the characteristic prodjections. Also for those
analogs the Weyl type inequalities are established, which reduce in various special
cases to the well-known inequalities [13�17]. Note that the de�ning properties of
Weyl solutions for the scalar Sturm�Liouville equation and the two-dimensional
Dirac system have been established by V.A. Marchenko [16]. The Weyl function
for abstract operators has been introduced and investigated in [18] (see also [19,
20]).
The proof of a great deal of our results involve the linear manifolds of the form
L = fA1f �A2f jf 2 Dg � H2; (0.2)
where Aj , j = 1; 2, are linear operators in H, D = DA1
= DA2
. These linear
manifolds are semi-de�nite in an inde�nite metric generated by an operator of
the form Q = diag(Q1;�Q2), with Qj = Q�j 2 B(H), Q�1
j 2 B(H); j = 1; 2. In
particular, the Q-nonnegativity for L means that
(Q1A1f;A1f)� (Q2A2f;A2f) � 0; 8f 2 D: (0.3)
In Section 2 we obtain a description for all such pairs of linear operators
A1, A2 that the linear manifold of the form L (0.2) is maximal Q-semi-de�nite.
(In the latter case such pair with dimH < 1, Q1 = Q2 = J = J�1 is called
J-nonstretching or J-stretching, nonsingular pair in terms of the J-theory by
V.P. Potapov. A di�erent description for such pair was found by S.A. Orlov [21]
using the J-theory). As a consequence we obtain a description of maximal Q-
semi-de�nite linear manifolds L of the form (0.2) in terms of the linear condition
that makes related the vectors A1f , A2f .
Note that (as it has been demonstrated in this work) the maximal Q-semi-
de�nite linear manifold L of the form (0.2) reduces to either a dissipative or
accumulative relation (of a special form) in H. In the general case a description
*A corrector's notice: also expounded in [12].
150 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On the Characteristic Operators and Projections and on the Solutions of Weyl...
of such relation is given in terms of its Cayley transform in [22] (see also [2, 3]).
On the contrary, our approach is based on describing maximal de�nite subspaces
in the space of M.G. Krein (see [23�25]).
In Section 2 we also produce necessary and su�cient conditions (Ths. 2.3, 2.7)
for the inequality (0.3) being separated, that is equivalent to the following two
inequalities being satis�ed simultaneously:
(Q1A1f;A1f) � 0; (Q2A2f;A2f) � 0:
It is just Th. 5.7 where generally unbounded idempotent (speci�c for dimH =1)
arises. In the case of L (0.2) corresponding to the equation (0.1) it becomes a
characteristic projection from B(H).
Also in Section 2 we consider the case when in (0.2) the operators Aj = Aj(�),
j = 1; 2, depend analytically on � in a domain � � C. In this case we obtain
a condition under which inequality (0.3) being separated for some � = �0 2 �
implies its separation for all � 2 �.
We denote the scalar products and norms in various spaces by (�; �) and k � k
(along with distinguishing indices if necessary). In view of its large volume, the
work splits into three parts. Part I is formed by Sect. 1, Part II � by Sect. 2, and
Part III � by Sects. 3, 4.
The Author would like to thank to F.S. Rofe-Beketov for his great attention
to this work.
1. Characteristic Operator. Its De�nition, Properties,
and Existence
Throughout the text we assume that for the operator-valued function H�(t) =
H�
��
(t) 2 B(H) in (0.1) there exists on I a conull set " such that:
1) There is such set A � CnR1 that every its point has a neighborhood
independent of t 2 ", in which H�(t) depends analytically on � at every �xed
t 2 ".
2) For all � 2 A, H�(t) is Bochner locally integrable in the uniform operator
topology (B-integrable).
3) The weight w�(t) = ImH�(t)=Im� is nonnegative (t 2 ", Im� 6= 0).
(Use the fact that H�(t) is Nevanlinna to show that for all � 2 ATR1, t 2 "
9u� lim
�!��i0
w�(t) = w�(t), with the operator-valued function w�(t) being locally
B-integrable).
Let X�(t) be the operator solution of (0.1) with f(t) = 0, normalized by the
condition X�(c) = I, where c 2 �I, I denotes the identity operator in H (in terms
of [24] X�(t) is the Cauchy operator of (0.1)). Introduce the notation Q(c) = G.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 151
V.I. Khrabustovsky
It follows from H�(t) = H�
��
(t) that with � 2 A one has
X�
��
(t)Q(t)X�(t) = G: (1.1)
For any �, � 2 �I, � � �, we use the notation��(�; �) =
�R
�
X�
�(t)w�(t)X�(t)dt,
N = fh 2 Hjh 2 Ker��(�; �); 8�; �g, and let P be the orthogonal projection
onto N?.
Lemma 1.1. 1o: N is independent of � 2 A. Additionally, if h 2 N , then
X�(t)h is a solution of
i
2
�
(Q(t)x(t))0 +Q(t)x0(t)
�
= (ReHi(t))x(t): (1.2)
2o: Suppose that for a subset F � H the following condition holds:
9�0 2 A; �; � 2 �I; number Æ > 0 : (��0(�; �)h; h) � Ækhk2; 8h 2 F: (1.3)
Let c 2 [�; �]. Then (1.3) is still valid if one replaces �0 with an arbitrary � 2 A
and Æ with some Æ(�) > 0.
P r o o f o f 1o. We are about to apply the following representation (to
deduce it, use, for example [26, p. 36]):
H�(t) = A(t) + �B(t) +
1Z
�1
�
1
� � �
� �
1 + �2
�
d�t(�); t 2 "; � 2 A: (1.4)
Here A(t) = ReHi(t), 0 � B(t) 2 B(H), for any �xed t 2 " the operator-valued
function �t(�) is nondecreasing and
1R
�1
d(�t(�)g;g)
1+�2
<1, g 2 H. Then
w�(t) = B(t) +
1Z
�1
d�t(�)
j� � �j2 ; t 2 "; � 2 A: (1.5)
Our subsequent argument requires
Proposition 1.1. Suppose that for g 2 H there exist t0 2 " and �0 2 A such
that w�0(t0)g = 0. Then for all � 2 A we have w�(t0)g = 0, h�(t0)g = 0 with
h�(t) = H�(t)�A(t).
152 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On the Characteristic Operators and Projections and on the Solutions of Weyl...
P r o o f o f P r o p o s i t i o n 1.1. Since the terms in (1.5) are
nonnegative and �t0(�) is constant in neighborhoods of points from A \ R
1 , it
follows from w�0(t0)g = 0 that B(t0)g = 0, and there exists some c > 0 such that
0 =
1R
�1
d(�t0 (�)g;g)
j���0j2 � c([�t0(�2) � �t0(�1)]g; g) with �1 < �1 < �2 < 1. Hence
d(�t0(�)g; g) = 0, which, together with (1.4) and (1.5), implies our statement.
Turn back to the proof of 1o. Let h 2 N at � = �0 2 A. By a virtue of
Prop. 1.1 for all � 2 A we have h�(t)X�0(t)h = 0 for a.a. t 2 I. This implies
that for all � 2 A, X�0(t)h is a solution of (1.2), and also a solution of (0.1)
with f(t) = 0. Hence there exists g 2 H such that X�0(t)h = X�(t)g for � 2 A.
Setting t = c, we deduce that h = g. Thus by Prop. 1.1, at a.a. t 2 I one has
w�(t)X�(t)h = w�(t)X�0(t)h = 0, that is, h 2 N for all � 2 A, so 1o is proved.
2o: For t 2 ", �; � 2 A, g 2 H, we have
H�(t)�H�(t)
�� �
g
� kB(t)gk +
0
@ 1Z
�1
d(�t(�)g; g)
j� � �j2
1
A
1=2
sup
h2H;khk=1
0
@ 1Z
�1
d(�t(�)h; h)
j� � �j2
1
A
1=2
;
whence, in view of (1:5), the following implication holds
(w�(t)gn; gn)! 0 ) H�(t)�H�(t)
�� �
gn ! 0; gn 2 H: (1.6)
Now assume there exist � 2 A and a sequence fhng 2 F with khnk = 1 such
that (��(�; �)hn; hn) ! 0. Hence fhng contains a subsequence fhnjg such that
at a.a. t 2 (�; �) one has
(w�(t)X�(t)hnj ;X�(t)hnj )! 0: (1.7)
Since the vector function xnj (t) = X�(t)hnj is a solution of the equation
i
2
�
(Q(t)x(t))0 +Q(t)x0(t)
��H�0(t)x(t) = (H�(t)�H�0(t)) x(t);
we have
xnj (t) = X�0(t)
2
4hnj +
tZ
c
X�1
�0
(s)(iQ(s))�1 [H�(s)�H�0(s)] xnj (s)ds
3
5 : (1.8)
On the other hand, xnj (t)! 0 in L2
w�(t)
(�; �), and the integral in (1:8) goes to
zero uniformly in t 2 [�; �] due to (1.7), (1.6) together with local B-integrability
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 153
V.I. Khrabustovsky
of H�(t) since c 2 [�; �]. Therefore X�0hnj ! 0 in L2
w�
(�; �) and hence in
L2
w�0
(�; �). This is because (1.5) implies that at any �xed t 2 " the norms
(w�(t)f; f)
1=2 are equivalent while � is within any compact K � A. Even more,
the constants in inequalities between the norms do not depend on t 2 "; � 2 K.
This contradicts (1.3). Lemma 1.1 is proved.
Lemma 1.1 with a constant leading coe�cient in (0.1) can be found in [1].
For x(t) 2 H or x(t) 2 B(H) introduce a notation U [x(t)] = (Q(t)x(t); x(t))
or U [x(t)] = x�(t)Q(t)x(t), respectively.
De�nition 1.1. An analytic operator-valued functionM(�) =M�(��) 2 B(H)
of nonreal � is called a characteristic operator (c.o.) of the equation (0.1) on I (or
brie�y c.o.), if for Im� 6= 0 and for any H-valued vector function f(t) 2 L2
w�
(I)
with compact support the corresponding solution x�(t) of (0.1) of the form
x�(t) � R�f =
Z
I
X�(t)
�
M(�)� 1
2
sgn(s� t)(iG)�1
�
X�
��
(s)w�(s)f(s)ds (1.9)
satis�es the condition
Im� lim
(�;�)"I
(U [x�(�)] � U [x�(�)]) � 0; Im� 6= 0: (1.10)
Note that (1.10) is equivalent to claiming that for any �nite [�; �] � suppf(t)
((�; �) � (a; b)) one has
Im�(U [x�(�)]� U [x�(�)]) � 0; Im� 6= 0; *
since the operator-valued function Im�X�
�(t)Q(t)X�(t) is nondecreasing for
Im� 6= 0. The latter is true because for any �nite (�; �) � (a; b), � 2 A, one has
X�
�(�)Q�(�)X�(�)�X�
�(�)Q�(�)X�(�) = 2Im���(�; �): (1.11)
The following remark provides a relationship between c.o.'s and regular bound-
ary problems for (0.1) with boundary conditions that depend on the spectral
parameter.
Remark 1.1 (Cf. [1]). Let I = (a; b) be a �nite interval and suppose (1.3) is
valid with F = H.
1o. If the operator-valued functions M�;N� depend analytically on nonreal �,
kM�hk+ kN�hk > 0; (0 6= h 2 H); (1.12)
*One can readily use this remark to deduce that (1.10) holds for anyH-valued vector function
f(t) 2 L2
w(I) with compact support, as it holds for any piecewise-continuous vector function
f(t) 2 H with compact support.
154 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On the Characteristic Operators and Projections and on the Solutions of Weyl...
and L� = fM�h � N�hjh 2 Hg � H2 are maximal Im�diag(Q(a);�Q(b))-
nonnegative subspaces in H2, (and so
Im�(N �
�Q(b)N� �M�
�Q(a)M�) � 0; Im� 6= 0); (1.13)
then the boundary problem given by attaching to (0.1) the boundary condition (in
the form of [4])
9h = h(�; f) 2 H : x(a) =M�h x(b) = N�h (1.14)
has for Im� 6= 0 a unique solution x�(t) as in (1.9), with
M(�) = �1
2
(X�1
� (a)M�+X
�1
� (b)N�)(X
�1
� (a)M��X�1
� (b)N�)
�1(iG)�1; (1.15)
and
(X�1
� (a)M� �X�1
� (b)N�)
�1 2 B(H): (1.16)
Here M(�) = M�(�) for �Im� > 0, with M+(�) and M�(�) being some
(di�erent in general) c.o.'s of (0.1) on I, so that
M+(�) =M�(�),M�
��
Q(a)M� = N �
��
Q(b)N�; Im� 6= 0: (1.17)
2 o. If M(�) is a c.o. of (0.1) on I then x�(t) is a solution of a boundary
problem as in 1o.
P r o o f o f 1o. Uniqueness of solution of the problem (0.1), (1.14), (1.13)
follows from (1.11),(1.13), Lemma 1.1 and the condition (1.3) with F = H.
Prove (1.16) assuming for certainty that Im� > 0. Denote
T� = X�1
� (a)M� �X�1
� (b)N�:
Prove that 0 =2 �p(T�) [ �c(T�). For if not, then for some � = �0 2 C+ there
exists a sequence of vectors ffng such that kfnk = 1, T�0fn ! 0, whence, with
the notation Y = X�0(b)X
�1
�0
(a), one has��N �
�0
Q(b)N�0 �M�
�0
Q(a)M�0
�
fn; fn)
�
+ ([Q(a)� Y �Q(b)Y ]M�0fn;M�0fn))
= (Q(b)N�0fn;N�0fn)� (Q(b)YM�0fn; YM�0fn)! 0: (1.18)
Thus each term in the left hand side of (1.18) goes to 0 since both are nonposi-
tive: the �rst one due to (1.13) and the second one due to (1.11). If so,M�0fn ! 0
due to (1.11) and the condition (1.3) with F = H, hence also N�0fn ! 0. This
implies that Sfn ! 0, where the operator S 2 B(H) corresponds to L�0 in view
of (2.4),(2.5). Since L�0 is a maximal Im�0diag(Q(a);�Q(b))-nonnegative sub-
space, R(S) = H by Theorem 2.1. Furthermore, KerS = f0g, as if for some
nonzero f 2 H Sf = 0, then by condition (2.6) of Th. 2.1 S1f = 0 )M�0f =
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 155
V.I. Khrabustovsky
N�0f = 0, which contradicts (1.12). Thus S�1 2 B(H) by the Banach theorem.
The contradiction we get proves that 0 =2 �p(T�) [ �c(T�).
Now prove that 0 =2 �r(T�). For if not, then there exist �0 2 C + and a nonzero
f 2 H with T ��0Gf = 0, whence by a virtue of (1.1)
0 = N �
�0
(�Q(b))Q�1(b)X��1
�0
(b)Gf +M�
�0
Q(a)Q�1(a)X��1
�0
(a)Gf
= �N �
�0
Q(b)X��0
(b)f +M�
�0
Q(a)X��0
(a)f:
Thus the vector X��0
(a)f � X��0
(b)f 2 H2 is diag(Q(a);�Q(b))-orthogonal
to L�0 , and hence is diag(Q(a);�Q(b))-nonpositive [25, p. 73]. On the other
hand by (1.11) it is diag(Q(a);�Q(b))-nonnegative. Therefore ���0
(a; b)f = 0 by
(1.11), whence f = 0 by Lemma 1.1 and the condition (1.3) with F = H. That
is 0 =2 �r(T�) and (1.16) is proved.
Now we are in a position to verify directly that (1.9), (1.15) is a solution of
the problem (0.1), (1.14), (1.13); with M�(�) being a c.o. by a virtue of (1.13)
and 50 of Theorem 1.1 (our proof of this theorem does not elaborate Remark 1.1).
It follows from (1.1) that for M(�) (1.15) one has
M(�) =M�(��),M�
��
Q(a)M� = N �
��
Q(b)N�; (1.19)
hence statement (1.17), and 1o is proved.
20. Let M(�) be a c.o. of (0.1). Represent M(�) in the form
M(�) =
�
P(�) � 1
2
I
�
(iG)�1: (1.20)
Then x�(t) (1.9),(1.20) is a solution of the problem (0.1), (1.14) with
M� = X�(a)(P(�) � I); N� = X�(b)P(�); (1.21)
so that M�, N� (1.21) obviously satisfy (1.12); also by 30 of Theorems 1.1,
2.4 and Lemma 2.6, L� related to M�, N� (1.21) is a maximal Im�diag(Q(a),
�Q(b))-nonnegative subspace. Remark 1.1 is proved completely.
Let �(t) be an operator solution of (Q(t)x(t))0 +Q(t)x0(t) = 0 normalized by
the condition �(c) = I. We have
�(t);��1(t) 2 B(H);�(t) 2 ACloc;�
�(t)Q(t)�(t) = G: (1.22)
Lemma 1.2. The substitution x(t) = �(t)y(t) reduces (0.1) to the equation
iGy0(t)� ~H�(t)y(t) = ~w�(t)g(t); t 2 �I; * (1.23)
*A di�erent substitution which reduces (0.1) to an equation with a constant leading coe�cient
is found in [27] (see also [3]).
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with Nevanlinna's operator function
~H�(t) = ��(t)H�(t)�(t) (1.24)
satisfying the same conditions as H�(t), and the weight
~w�(t) = ��(t)w�(t)�(t); (1.25)
g(t) = ��1(t)f(t).
The equations (0.1) and (1.22)�(1.25) have the same c.o.'s.*
P r o o f. The relation (1.23) for y(t) = ��1(t)x(t) allows a direct veri�cation.
Let B(H) 3 M(�) be an arbitrary operator-valued function, x�(t) (1.9) the
associated solution of (0.1), y�(t) = ��1(t)x�(t). It is easy to see that y�(t)
is also given by (1.9) after substituting therein X�(t) by the Cauchy operator
for the equation (1.23) Y�(t) = ��1(t)X�(t), the weight w�(t) by ~w�(t) (1.25),
and the vector function f(t) by g(t) = ��1(t)f(t). Thus by a virtue of (1.22)
(Q(t)x�(t); x�(t)) = (Gy�(t); y�(t)) for t 2 �I. It follows that if M(�) is a c.o. of
(0.1), then M(�) is a c.o. of (1.22)�(1.25), and the converse is also true. The
Lemma is proved.
Lemma 1.3. Let Aj 2 B(H), j = 1; 2. Then: I.
(A1 �A2)
�1 2 B(H); (1.26)
if either of the following conditions holds:
1o.
(�1)jA�jGAj � 0; j = 1; 2; (1.27)
the image R(A1) or R(A2) is uniformly G-de�nite,
L = fA1f �A2f jf 2 Hg � H2 (1.28)
is a maximal G2 = diag(G;�G)-nonnegative subspace in H2, and
kA1fk+ kA2fk > 0; 0 6= f 2 H: (1.29)
2o. There exists a positive constant Æ > 0 such that either
a)
A�1GA1 � 0; (I �A�1)G(I �A1) � �Æ(I �A�1)(I �A1); (1.30)
A�2GA2 � �ÆA�2A2; (I �A�2)G(I �A2) � 0; ** (1.31)
*And, obviously, the same operators ��(�; �).
**It is demonstrated in �2 that (1.30), (1.31) imply (even with Æ = 0) that A1 and A2 are
projections.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 157
V.I. Khrabustovsky
or
b)
A�1GA1 � 0; (I �A�1)G(I �A1) � 0; (1.32)
A�2GA2 � �ÆA�2A2; (I �A�2)G(I �A2) � Æ(I �A�2)(I �A2): (1.33)
II. If Bj 2 H, j = 1; 2, then
(A1B1 �A2B2)
�1 2 B(H); (1.34)
when
A�1GA1 � 0; (I �A�1)G(I �A1) � 0; (1.35)
and there exists such positive constant Æ > 0 that
A�2GA2 � �ÆA�2A2; (I �A�2)G(I �A2) � 0; (1.36)
B�
1GB1 � ÆB�
1B1; B�
2GB2 � �ÆB�
2B2; (1.37)
fB1f �B2f jf 2 Hg � H2 (1.38)
is a maximal G2-nonnegative subspace in H2, and
kB1fk+ kB2fk > 0; 0 6= f 2 H: (1.39)
P r o o f. I.1o. Prove (1.26) e.g., for A1 + A2. Suppose for certainty that
R(A1) is uniformly G-de�nite, i.e.
9Æ > 0 : A�1GA1 � ÆA�1A1: (1.40)
Prove that
0 =2 �p(A1 +A2) [ �c(A1 +A2): (1.41)
If one assumes the contrary, then
9ffng; fn 2 H; kfnk = 1 : (A1 +A2)fn ! 0; (1.42)
whence
(A�1GA1fn; fn) + (�(A�2GA2fn; fn))! 0: (1.43)
It follows that each term in the left hand side of (1.43) goes to 0, as both are
nonnegative due to (1.27). Then by a virtue of (1.40), (1.42) one has
A1fn ! 0; A2fn ! 0: (1.44)
Therefore Sfn ! 0, with S 2 B(H) being associated to L (1.28) as in (2.4), (2.5).
Since L (1.28) is maximal, R(S) = H by Th. 2.1. Besides that, KerS = f0g, as if
Sf = 0 for some nonzero f 2 H, then by the condition (2.6) of Th. 2.1 S1f = 0,
158 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
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hence A1f = A2f = 0, which contradicts (1.29). Therefore the Banach theorem
implies S�1 2 B(H). The contradiction we get proves (1.41).
Now prove that 0 =2 �r(A1 +A2). If not, then
90 6= f 2 H : (A�1 +A�2)f = 0; (1.45)
whence
8h 2 H (GG�1f;A1h) + (GG�1f;A2h) = 0: (1.46)
(1.46) means that
(G�1f � (�G�1f)) 2 L[G2];
where [A] stands for the A-orthogonal complement in the associated Hilbert space,
whence by (1.27) and maximality of L (1.28) we deduce by Lemma 2.7 that
G�1f 2 fA1hjh 2 Hg[G]; G�1f 2 fA2hjh 2 Hg[G]: (1.47)
In view of [25, p. 74] the �rst statement in (1.47), together with (1.40) implies
that
9Æ1 > 0 : (GG�1f;G�1f) � �Æ1kG�1fk2; (1.48)
and the second statement in (1.47) implies that
(GG�1f;G�1f) � 0: (1.49)
It follows from (1.48), (1.49) that f = 0, so that I.1o is proved.
I.2o. Assume for certainty that a) holds.
Prove (1.41). If (1.41) fails, then (1.42) holds, and hence (1.43) holds as well.
Use the initial inequalities in (1.30), (1.31) to deduce from (1.43) that
(I �A1)fn � fn ! 0; (I �A2)fn � fn ! 0; (1.50)
in the same way as in the proof of (1.44), whence
((I �A�1)G(I �A1)fn; fn)) + (�(I �A�2)G(I �A2)fn; fn))! 0: (1.51)
Now use the �nal inequalities in (1.30), (1.31) to deduce that (I �A1)fn ! 0
in the same way as in the proof of (1.44); this, together with (1.50) implies fn ! 0,
which is impossible. Thus (1.41) is proved.
Now prove 0 =2 �r(A1 +A2). If this fails, then (1.45) holds, whence
(G�1A�1f;A
�
1f) = (G�1A�2f;A
�
2f): (1.52)
On the other hand, by Lemma 2.4 and Ths. 2.4, 2.7 one has A2
1 = A1 in view
of (1.30). Therefore the vector G�1A�1f is G-orthogonal to f(I�A1)hjh 2 Hg, the
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 159
V.I. Khrabustovsky
latter subspace being in view of the �nal inequality in (1.30) maximal uniformly
G-negative by theorem 2.4. Thus by [25, p. 74]
9Æ1 > 0 : (G�1A�1f;A
�
1f) � Æ1kG�1A�1fk2:
In a similar way deduce that the right hand side of (1.52) is nonpositive, whence
A�1f = A�2f = 0: (1.53)
But in this case (1.47) holds, and our proof can be completed in the same way as
that of I.1o, if one applies the initial inequalities in (1.30), (1.31).
II. Prove e.g., that (A1B1 +A2B2)
�1 2 B(H). Demonstrate �rst that
0 =2 �p(A1B1 +A2B2) [ �c(A1B1 +A2B2): (1.54)
If this fails, then
9ffng; fn 2 H; kfnk = 1 : (A1B1 +A2B2)fn ! 0: (1.55)
Use the initial inequalities in (1.35), (1.36), to deduce from (1.55) that
(I �A1)B1fn �B1fn ! 0; (I �A2)B2fn �B2fn ! 0; (1.56)
in the same way as in the proof of (1.44), whence
� ((I �A�1)G(I �A1)B1fn; B1fn) + (B�
1GB1fn; fn)! 0; (1.57)
((I �A�2)G(I �A2)B2fn; B2fn) + (�(B�
2GB2fn; fn))! 0: (1.58)
Each term in (1.57) goes to zero, as they are nonnegative by the �nal inequality
in (1.35) and the initial inequality in (1.37). Then in view of (1.37) one has
B1fn ! 0: (1.59)
In a similar way, it follows from (1.58), (1.36), (1.37) that
B2fn ! 0: (1.60)
In the same way as in the proof of I.1o one can demonstrate that (1.59), (1.60)
contradicts the maximality condition for the subspace (1.38) and the condition
(1.39). This proves (1.54).
Now prove that 0 =2 �r(A1B1 +A2B2). If this fails, then
90 6= f 2 H : (B�
1A
�
1 +B�
2A
�
2)f = 0: (1.61)
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Use the maximality of (1.38), (1.37) deduce from (1.61) by Lemma 2.7 that
G�1A�1f 2 fB1hjh 2 Hg[G]; G�1A�2f 2 fB2hjh 2 Hg[G];
in the same way as in the proof of I.1o. From this we deduce again (1.53) by
analogy with the proof of I.1o and in view of (1.37). The proof �nishes in the
same way as that of I.1o if one uses initial inequalities in (1.35), (1.36). The
lemma is proved.
Note that in I.1o of Lemma 1.3 one could not get rid of niether the condition
of uniform G-de�niteness for R(A1) or R(A2), nor the maximality condition for
L (1.28).
In fact, (1.26) is not valid for
G =
�
0 i
�i 0
�
; A1 =
�
1 0
1 0
�
; A2 =
�
0 1
0 1
�
;
although all the assumptions of I.1o hold except the uniform G-de�niteness for
R(A1) or R(A2).
Also for H = C � l2
G = diag(�1; 1; 1; : : :); A1 = diag(0; U); A2 = (1; 0; 0; : : :);
with U being the one-sided shift in l2 (see [28]), (1.26) does not hold, although
all assumptions of I.1o but the maximality of L (1.28) are satis�ed.
Note also that (1.26) fails if only (1.35), (1.36) are valid (and even R(A1) is
uniformly G-de�nite). To see this, one should set
G =
�
0 i
�i 0
�
; A1 =
�
1 0
�i 0
�
; A2 =
�
1 0
i 0
�
:
Thus introducing operators Bj 6= I is necessary to make sure (1.34) is valid.
In view of [1] Lemma 1.3 provides an explicit expression for the projection
onto R(A1) parallel to R(A2) in terms of A1 and A2.
Example 1.1. Suppose I = (a; b) is a �nite interval, a < c < b, and the
condition (1.3) with F = H holds for I = I� = (a; c) and for I = I+ =
(c; b). Introduce the notation Y�(t) = ��1(t)X�(t), with �(t) being as in (1.22).
Then Y�(a) and Y�(b) are uniform �G-compressions and uniform �G-stretchings
respectively as �Im� > 0 due to (1.11), while Y �
� (a) and Y �
� (b) are uniform
�G�1-compressions and �G�1-stretchings respectively due to (1.1), (1.11) as
�Im� > 0. Therefore [24, p. 66] with G inde�nite, the operators Y�(a) and
Y�(b) are unitarily dichotomic as Im� 6= 0. Denote by P(Y�(a)) and P(Y�(b))
the Riesz projections for Y�(a) and Y�(b) corresponding to those parts of their
spectra which are inside the unit circle. Let
H�(�) = P(Y�(a))H; H+(�) = P(Y�(b))H: (1.62)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 161
V.I. Khrabustovsky
Since the subspaces H�(�); (I � P(Y�(b)))H and H+(�); (I � P(Y�(a)))H are
respectively uniformly �G-positive and uniformly �G-positive as �Im� > 0 by
[24, p. 64], it follows from [25, p. 76] that
H = H�(�)uH+(�): (1.63)
Denote by P(�) the projection onto H+(�) parallel to H�(�). By I.2o of Lem-
ma 1.3, (P(Y�(b)) + P(Y�(a)))�1 2 B(H), and hence by [1] (see also Sect. 2 of
this work)
P(�) = P(Y�(b))(P(Y�(b)) + P(Y�(a)))�1; (1.64)
which implies that P(�) depends analytically on nonreal �. Now it is easy to see
in view (1.1), (1.19), (1.22) that the operator-valued function M(�) (1.20) with
P as in (1.64), is a c.o. for (0.1) which corresponds (in terms of Remark 1.1) to
boundary conditions like (1.14) with
M� = ��(a)P(Y�(a))Y�(a); N� = �(b)P(Y�(b))Y�(b):
Theorem 1.1o. Condition (1.10) for a the solution (1.9) as in the de�nition
of c.o. is equivalent to the following inequality:
kR�fk2L2
w�
(I) � Im(R�f; f)L2
w�
(I)=Im�; Im� 6= 0: (1.65)
2o. Condition (1.10) for an operator-valued function M(�) 2 B(H) of the
form (1.20) and arbitrary H-valued vector functions f(t) 2 L2
w�
(I) with compact
support is equivalent to the following condition:
8[�; �] � �I : Im�(�+
� (�)� ��� (�)) � 0; Im� 6= 0; (1.66)
with
�+
� (t) = U [X�(t)PG�1P ]; ��� (t) = U [X�(t)(I �P)G�1P ]: (1.67)
3o. Condition (1.10) for an operator-valued function M(�) 2 B(H) of the
form (1.20) and arbitrary H-valued vector functions f(t) 2 L2
w�
(I) with compact
support is equivalent to the following condition
PG�1 [(I �P�(�))��(�; c)(I �P(�)) + P�(�)��(c; �)P�(�)]G�1P
� Im [PM(�)P ]
Im�
; Im� 6= 0 (1.68)
for any �nite � � c � �, [�; �] � �I. Hence for any c.o. M(�):
Im [PM(�)P ]
Im�
� 0; Im� 6= 0: (1.69)
162 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
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4o. If M(�) is a c.o. for (0.1), and an analytic operator-valued function of
nonreal � M0(�) = M�
0 (
��) 2 B(H), PM0(�)P = 0, then M(�) +M0(�) is also
a c.o., hence PM(�)P is a c.o as well.
5o. The validity of condition (1.10) for an operator-valued function M(�) 2
B(H) and arbitrary H-valued vector functions f(t) 2 L2
w�
(I) with compact support
in one of complex half-planes implies its validity for any such f(t) in another half-
planes, if one sets up M(��) =M�(�) in (1.9).
6o. A c.o. in general is not uniquely determined by the problem (0.1), (1.10)
(in particular for P = I); however, if Mj(�) are c.o.'s (j = 1; 2), and there exists
a nonreal � such that if f 2 H and lim(�;�)"I(��(�; �)f; f) <1 implies f 2 N ,
then P [M1(�)�M2(�)]P = 0. If one assumes additionally (1.3) with F = H then
M1(�) =M2(�), for all � 2 CnR1.
P r o o f o f 1o. To see that (1.10) and (1.65) are equivalent, note that for
any solution x(t) of (0.1) and any [�; �] � �I one has for Im� 6= 0
kx(t)k2L2
w�
(�;�) �
Im(x(t); f(t))L2
w�
(�;�)
Im�
=
U [x(�)]� U [x(�)]
2Im�
: (1.70)
To prove 2o and other statements we need
Lemma 1.4. Denote by F the set of H-valued vector functions f(t) 2 L2
w�
(I)
with compact supports suppf(t) � �I, and de�ne an operator
I�f =
Z
I
X�
��
(s)w�(s)f(s)ds; (1.71)
which maps F into H.
1) Let [�; �] � �I. Then
1) 8� 2 A : ���(�; �)H � I�F � N?; (1.72)
and
8� 2 A : I�F = N?: (1.73)
2) Suppose there exist �0 2 A and �0; �0 2 �I (�0 � c � �0) such that
Ker��0(�0; �0) = N . Then
8� 2 A : ��(�0; �0)H = N?: (1.74)
P r o o f o f L e m m a 1.4. 1) Setting for all � 2 A, h 2 H
f(t) =
(
X��(t)h; t 2 [�; �]
0; t =2 [�; �]
; (1.75)
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V.I. Khrabustovsky
we obtain I�f = ���(�; �)h, and the left inclusion in (1.72) is proved. The right
inclusion in (1.72) can be deduced by observing that if h 2 N then for all � 2 A,
f(t) 2 F one has (I�f; h) =
R
I
(f(t); w�(t)X��(t)h)dt = 0, since w�(t)X��(t)h = 0
(a.e.) in view of the de�nition of N and Lemma 1.1.
If (1.73) fails for some � = �0 2 A, then by (1.72) there exists a nonzero
g 2 N? such that for all f(t) 2 F one has (I�0f; g) = 0. Substituting f(t) as in
(1.75) with � = �0, h = g, we obtain
�
���0
(�; �)g; g
�
= 0 for all �; � 2 �I, whence
g 2 N by Lemma 1.1. 1) is proved.
2) Follows from Lemma 1.1. Lemma 1.4 is proved.
Turn back to the proof of 2o of Theorem 1.1. Fix a nonreal �. Let (�j ; �j) =
Ij " I, with the �nite intervals Ij being such that c 2 �Ij. Introduce Nj =
Ker��(�j ; �j) and denote by Qj the orthogonal projection onto Nj . As an non-
increasing sequence of orthogonal projections, Qj has a strong limit Q, which is
again an orthogonal projection. Prove that Q projects onto N .
In fact, let Qh = h for some h 2 H. Then Nj 3 Qjh! Qh = h. On the other
hand, Nj � Nk for k � j, whence Qjh 2 Nk for k � j and therefore h 2 Nk for
all k, which implies h 2 \kNk = N .
Let M(�)(1:20) 2 B(H) satis�es (1.10) and f 2 N?. Introduce the notation
fj = Qjf , gj = (I � Qj)f . One has f = fj � gj , where fj ! 0. On the other
hand, by 2) of Lemma 1.4 for every gj there exists a sequence fhjng 2 H such that
���(�j ; �j)h
j
n ! gj . Hence in view of (1.11), when Ij � (�; �) :
Im�
�
U
�
X�(�)PG�1gj
�� U
�
X�(�)(I �P)G�1gj
�
� lim
n!1
Im� fU [x�(�j)]� U [x�(�j)]g ; (1.76)
where x�(t) is determined by (1.9) with f(t) being as in (1.75) with h = h
j
n,
[�; �] = [�j ; �j ]. In view of (1.10), the left hand side of (1.76) is nonpositive for
Im� 6= 0, whence by (1.76) Im�
�
U
�
X�(�)PG�1f
�� U
�
X�(�)(I �P)G�1f
�
� 0 for all f 2 N?. Now (1.66) is proved for M(�).
Conversely, suppose that an operator M(�) 2 B(H) as in (1.20) satisfy (1.66).
Then the corresponding solutions x�(t) (1.9) satisfy (1.10) by 1) of Lemma 1.4,
which proves 2o.
3o follows from the fact that for all M(�)(1:20) 2 B(H) one has
(�+
� (�)� ��� (�)) + 2Im[PM(�)P ]
= 2Im�PG�1[(I �P�(�))��(�; c)(I �P)) + P�(�)��(c; �)P(�)]G�1P; (1.77)
together with 20.
4o. Take into account that by (1.11) X�
�(�)Q(�)X�(�)h0 = X�
�(�)Q(�)
164 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On the Characteristic Operators and Projections and on the Solutions of Weyl...
�X�(�)h0, for all h0 2 N it is easy to verify that
U [X�(�)(P(�)G�1Pf + (I � P )g)] � U [X�(�)((P(�) � I)G�1Pf + (I � P )g)]
= (�+
� (�)f; f)� (��� (�)f; f); 8f; g 2 H: (1.78)
Now represent M(�) in the form (1.20), and M(�) + M0(�) in the form
M(�) +M0(�) =
�P(�) + �P(�)� 1
2
I
�
(iG)�1, then M0 = �P(�)(iG)�1 hence
P�P(�)(iG)�1P = 0, so 4o follows from (1.78) and 20.
5o. Suppose for certainty that for Im� > 0, M(�) 2 B(H) satis�es (1.10),
and �; � 2 �I, � � c � �. Denote by P1 an analogue for the orthogonal projection
P with respect to I1 = (�; �) and represent M1(�) = P1M(�)P1 in the form
M1(�) =
�
P1(�)�
1
2
I
�
(iG)�1: (1.79)
Then by (1.11), 30 of Th. 1.1, Th. 2.4 and Lem. 2.6
L� =
�
X�(�)
�
(P1(�)� I)(iG)�1P1 + I � P1
�
f �X�(�)
� �P1(�)(iG)
�1P1 + I � P1
�
f jf 2 H (1.80)
is a maximal diag(Q(�);�Q(�))-nonnegative subspace. Therefore by Theorem
2.5, L�� determined by (1.80) after substituting P1(�) with P1(��) =
def G�1(I �
P�1 )(�))G is diag(Q(�);�Q(�))-nonpositive.
Therefore by Lemma 1.1, (1.78), 20 of Theorem 1.1, (1.10) is valid for Im� < 0
if one replaces in (1.10) I = I1, and also replaces in (1.9) M(�) with M�
1 (�) =
P1M
�(�)P1, hence even with M�(�) by 40 of Theorem 1.1. Thus 5o is proved
for any such H-valued vector function f(t) 2 L2
w�
(I) with compact support that
suppf(t) � [�; �], and hence due to arbitrariness of �; � for any H-valued f(t) 2
L2
w�
(I) with compact support as well.
60. LetM1(�) andM2(�) be c.o.'s. Denote by R
1
�f and R2
�f the corresponding
solutions of (0.1) of the form (1.9). By a virtue of (1.65) for c.o. that has already
been proved, one has R1
�0f � R2
�0f = X�0(t) [M1(�0)�M2(�0)] I�0f 2 L2
w�0
(I),
with I�f (see (1.71)). So the initial statement of 6o follows from (1.73).
Now the �nal statement follows from M(��) = M�(�), together with the fol-
lowing lemma, which could also make an independent interest.
Lemma 1.5. Suppose (1.3) holds with F = H, and for some � 2 CnR1 one
has
h 2 H; lim
(�;�)"I
(��(�; �)h; h) <1) h = 0: (1.81)
Then (1.81) also holds with � being replaced by any such � that Im�Im� > 0.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 165
V.I. Khrabustovsky
P r o o f o f L e m m a 1.5. Suppose there exists �0 2 CnR1 (�0 6= �,
Im�0Im� > 0) such that the equation l[y] = H�0(t)y has a solution 0 6� y(t) 2
L2
w�0
(I) (and hence y(t) 2 L2
w�
(I); � 2 A), with l[y] = i
2
((Q(t)y)0 + Q(t)y0).
Consider the nonhomogeneous equation
l[z]�H�(t)z = (H�0(t)�H�(t))y: (1.82)
Every its solution can be represented in the form
z(t) = y(t) +X�(t)g; g 2 H; (1.83)
so that z(t) =2 L2
w�(I) for g 6= 0.
Let M(�) be a c.o. for (0.1) (a proof of its existence does not elaborate of
Th. 1.1).
Use property (1.65) of c.o., (1.3) with F = H, and the functions of the form
(1.75) to demonstrate that there exist constants cj(�; �) such that
8� 2 C nR1 ; h 2 H : kX�(t)(P(�) � I)hkL2
w�
(a;�) � c1(�; �)khk;
kX�(t)P(�)hkL2
w�
(�;b) � c2(�; �)khk
with � 2 �I, P(�) (see (1.20)).
It follows that there exists a constant k(s; �) such thatZ
I
(w�(t)K(t; s; �)h;K(t; s; �)h)dt � k(s; �)khk2 (1.84)
with
K(t; s; �) = X�(t)
�
M(�) � 1
2
sgn(s� t)(iG)�1
�
X�
��
(s): (1.85)
It follows from (1.84) and representation (1.5) for the weight w�(t) (cf. the
proof of 20, Prop. 1.1), that the integral
x(t) =
Z
I
K(t; s; �)f(s)ds (1.86)
converges strongly, with f(t) = (H�0(t) � H�(t))v(t), where measurable v(t) 2
L2
w�(I).
Clearly, x(t) (1.86) is a solution of (1.82) with y(t) being replaced by v(t).
In the case when v(t) has a compact support suppf � [�; �] � �I, an argument
similar to that of the proof of 1o demonstrates
kx(t)k2Lw� (�;�) �
Im
�R
�
(x(t); f(t))dt
Im�
166 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On the Characteristic Operators and Projections and on the Solutions of Weyl...
since M(�) is a c.o.
One deduces that in general case x(t) 2 L2
w�(I) (hence in view of (1.83)
x(t) = y(t) when v(t) = y(t)). Now it is easy to demonstrate that there exists a
constant c such that
kx(t)� xn(t)kL2
w�(I)
� ckv(t) � vn(t)kL2
w�(I)
;
where vn(t) = �n(t)v(t), �n(t) are characteristic functions of �nite intervals
(�n; �n) " I, xn(t) the associated solutions of the form (1.86). On the other
hand
1
2
���(Q(t)x(t); x(t)) j�� � (Q(t)xn(t); xn(t)) j��
���
� jIm�j
���kx(t)k2L2
w�
(�;�) � kxn(t)k2L2
w�
(�;�)
���
+
����Im
Z �
�
[(x(t); f(t)) � (xn(t); fn(t))] dt
����! 0
uniformly in �, � 2 �I. Furthermore,
Im� (Q(t)xn(t); xn(t)) j�n�n � 0
since M(�) is a c.o. Hence
lim
(�n;�n)"I
Im� (Q(t)x(t); x(t)) j�n�n � 0
(the limit exists since x(t); v(t) 2 L2
w�(I)). In particular,
lim
(�n;�n)"I
Im� (Q(t)y(t); y(t)) j�n�n � 0: (1.87)
On the other hand by (1.11), one has
ky(t)k2L2
w�0
(�;�) =
(Q(t)y(t); y(t))j��
2Im�0
;
whence ky(t)kL2
w�
(I) = 0 in view of (1.87) and Im�0Im� > 0. Thus y(t) � 0,
which is due to (1.3) with F = H. Now Lemma 1.5, together with 6o of Th. 1.1,
are proved.
Conditions of existence for c.o. are given by
Theorem 1.2. A c.o. of (0.1) on I exists, if either one of the ends of I is
�nite or if for some �0 2 A \ R1 the norm kX�
�0
(t)w�0(t)X�0(t)k is summable at
one of the ends of I. Also, a c.o. exists if (1.3) holds with F = N?.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 167
V.I. Khrabustovsky
To prove Theorem 1.2, we need the following lemma, which could possibly
make an independent interest.
Lemma 1.6. Suppose that operator-valued functions Mn(�) are c.o. for (0.1)
on �nite or in�nite intervals In such that c 2 �In, In " I. Assume also that for
any compact K � CnR1
9c(K) : 8� 2 K kMn(�)k < c(K): (1.88)
Then there is a subsequence fnjg such that for Im� 6= 0 the limit
w � limMnj (�) =M(�) (1.89)
exists and M(�) is a c.o. for (0.1) on I.
P r o o f o f L e m m a 1.6. One can deduce from Vitali's theorem [29]
and weak compactness of the family Mn(�) at any �xed nonreal � (which is itself
due to (1.88)) that there exists a subsequence fnjg such that (1.89) holds, with
the operator-valued function M(�) =M�(��) being analytic for nonreal �.
Prove that M(�) is a c.o. for (0.1) on I. By (1.77) (which is valid for all
M(�)(1.20)2 B(H)) one has
8[�; �] � �I; � � c � �; f 2 H (�+
� (�)f; f)� (��� (�)f; f)
Im�
= 2[(��(�; c)(I �P(�))G�1Pf; (I �P(�))G�1Pf)
+(��(c; �)P(�)G�1Pf;P(�)G�1Pf)� Im(PMPf; f)
Im�
]: (1.90)
Denote by Pn an analog with respect to In for the orthogonal projection P
and by Pn(�) an analog P(�) (1.20) for Mn(�). Use nonnegativity of ��(s; t),
[30, pp. 176, 193], and the fact that Pn !s P to deduce from (1.90):
8[�; �] � �I; � � c � �; f 2 H :
(�+
� (�)f; f)� (��� (�)f; f)
Im�
� 2
�
lim(��(�; c)(I �Pn(�))G�1Pnf; (I �Pn(�))G�1Pnf)
+lim(��(c; �)Pn(�)G�1Pnf;Pn(�)G�1Pnf)� lim
Im(PnMn(�)Pnf; f)
Im�
�
� 2lim
�
(��(�; c)(I �Pn(�))G�1Pnf; (I �Pn(�))G�1Pnf)
+(��(c; �)Pn(�)G�1Pnf;Pn(�)G�1Pnf)�
Im(PnMn(�)Pnf; f)
Im�
� � 0 (1.91)
168 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On the Characteristic Operators and Projections and on the Solutions of Weyl...
since by 30 of Th. 1.1 the expression in brackets is nonpositive because Mn(�)
is a c.o. on the interval In, which contains (�; �) for n large enough. Thus for
M(�) (1.89), (1.66) holds if [�; �] � �I and, hence, if [�; �] � �I (in view, that
��� (t) depends on t continuously). Now by 2o of Th. 1.1, Lemma 1.6 is proved.
P r o o f o f T h e o r e m 1.2. Let G is de�nite. Then by 2o of Theorem 1.1
M(�) (1.20) with P = 1
2
(I � sgn(Im�G)) is a c.o. of (0.1). Let G is inde�nite.
By Lemma 1.2, it su�ces to prove the theorem for (0.1) with constant
Q(t) = G: (1.92)
Case I. Suppose one of the ends of I is �nite, e.g. a > �1. Consider (0.1),
(1.92) with the operator-valued coe�cient H�(t) being replaced by
Hn
� (t) =
(
H�; a < t < �n = minfb; ng
�I; �n < t <1;
(1.93)
with n � c. Then the Cauchy operator X�(t) is to be replaced by
Xn
� (t) =
(
X�(t); a < t < �n
e(iG)�1�(t��n)X�(�n); �n � t <1 : (1.94)
Using an argument similar to that use with Y�(b) in Ex. 1.1, we observe that for
Im� 6= 0, t > �n, the operators Y
n
� (t) = Xn
� (t)X
�1
� (a) are unitarily dichotomic.
Consider the subspaces
Hn
+(�) = P(Y n
� (�n + 1))H
similar to H+(�) in Ex. 1.1. Consider one more projection-valued function
�(�) =
(
�; Im� > 0;
I ��; Im� < 0;
(1.95)
where � project onto �xed maximal uniformly G-positive subspace parallel to its
G-orthogonal complement. The latter subspace is maximal uniformly G-negative
by [25, p. 74].
Therefore in view of [25, p. 76] one has
H = �(�)H uHn
+(�):
Denote by Pn(�) the projection onto X�1
� (a)Hn
+(�) parallel to X
�1
� (a)�(�)H. By
Lemma 1.3
(P (Y n
� (�n + 1)) + �(�))�1 2 B(H):
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 169
V.I. Khrabustovsky
Therefore by [1] (see also Sect. 2)
Pn(�) = X�1
� (a)P (Y n
� (�n + 1)) (P (Y n
� (�n + 1)) + �(�))
�1
X�(a) (1.96)
depends analytically on nonreal �. It is easy to see that in view of (1.95), (1.1),
(1.19)
Mn(�) =
�
Pn(�)�
1
2
�
(iG)�1 (1.97)
is a c.o. for (0.1), (1.92) with H�(t) = Hn
� (t) (1.93) on (a; �n + 1), hence also for
(0.1), (1.92) on I = In = (a; �n). Since n is arbitrary, the theorem is proved for
a �nite interval (a; b).
Proceed with proving the theorem for the case b = 1. Due to the uniform
�G-positivity of �(�)H and the uniform �G-negativity of Hn
+(�) for Im� 6= 0,
the mutual inclination (see [24]) of these subspaces at any �xed nonreal � is
separated out from zero uniformly in n. Therefore [24, p. 224] for any compact
K � C nR1 one has
9C(K) : 8� 2 K kPn(�)k � C(K): (1.98)
Now the statement of the theorem in the Case I follows from (1.98), (1.97),
and Lemma 1.6.
Case II. Assume, for example, a = �1,
9�0 2 A \R1 :
Z c
�1
X�
�0
(t)w�0(t)X�0(t)
dt <1: (1.99)
Lemma 1.7. If (1.99) holds, then the substitution x(t) = X�0(t)z(t) reduces
equation (0.1) with f(t) = 0 to the equation
iGz0(t)�X�
�0
(t) (H�(t)�H�0(t))X�0(t)z(t) = 0; t 2 �I: (1.100)
The equation (0.1) and its analog for (1.100) have the same c.o.'s. Also, for the
Cauchy operator Z�(t) = X�1
�0
(t)X�(t) of the equation (1.100)
9u� lim
t!�1
Z�(t) = Z�(1);
where Z�(1) depends analytically on � 2 A and Z�1
� (1) 2 B(H).
P r o o f o f L e m m a 1.7. The proof of coincidence of c.o. for (0.1) and
(1.100) is the same as that of Lemma 1.2.
An estimate for
X�
�0
(t) (H�(t)�H�0(t))X�0(t)
, like that preceding (1.6)
demonstrates in view of (1.5), (1.99):
cZ
�1
X�
�0
(t) (H�(t)�H�0(t))X�0(t)
dt <1;
170 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On the Characteristic Operators and Projections and on the Solutions of Weyl...
whence by [24, p. 166], [31] the statement of the lemma with respect to Z�(t)
follows. Lemma 1.7 is proved.
Now in view of Lemma 1.7, c.o. for (1.100) (hence also c.o. for (0.1)) can be
produced similarly to the Case I with X�(a) being replaced by Z�(1).
Case III. Suppose (1.3) holds for F = N?.
Lemma 1.8. Suppose that (1.3) holds for F = N? and in (1.3) either (�; �) �
(a; c) or (�; �) � (c; b). Let M 0(�) be a c.o. for (0.1) on the interval I 0 � I that
contains, respectively, either (�; c) or (c; �). Then for any compact K � C nR1
there exists a constant c = c(K) independent of M 0(�) and I 0 and such that
8� 2 K :
PM 0(�)P
� c(K):
P r o o f of the lemma follows from 3o of Th. 1.1.
In view of the Cases I and II of the theorem that have already been proved,
it su�ces to prove Case III for I = (�1;1).
First assume c =2 (�; �), e.g. for certainty c � �. Hence (1.3) remains valid
if one replaces therein (�; �) with (c; �). Similarly to (1.93) and (1.94), extend
H�(t) and X�(t) from (�n; n) � (c; �) onto (�n� 1; n+1) and use the argument
of Ex. 1.1 and Case I to obtain
H = P (Xn
� (�n� 1))Hu P (Xn
� (n+ 1))H:
Denote by Pn(�) the projection onto P (Xn
� (n+ 1))H parallel to
P (Xn
� (�n� 1))H. Similarly to Ex. 1.1 and Case I
Pn(�) = P (Xn
� (n+ 1)) (P (Xn
� (n+ 1)) + P (Xn
� (�n� 1)))�1 ; (1.101)
with (:::)�1 2 B(H).
It is easy to see that, similarly to Case I, Mn (1.20), (1.101) is a c.o. for (0.1),
(1.92) on (�n; n). By 4o of Th. 1.1 PMn(�)P is also a c.o. for (0.1) on (�n; n).
Now the proof for the case c =2 (�; �) follows from Lemmas 1.8, 1.6.
Finally, let c 2 (�; �). It is easy to see that by (1.3) with F = H? one has for
� 2 A:
9Æ1 = Æ1(�) > 0 : (X�
�
�1(�)��(�; �)X
�1
� (�)f; f) � Æ1kfk2; 8f 2 [X�(�)N ]?
(X�(�)N does not depend on � 2 A by Lem. 1.1).
Thus it follows from the above observations that there exists a c.o. ~M(�)
for (0.1) if its Cauchy operator is normalized as the identity not at c but at �
(i.e. X�(t) is replaced by X�(t)X
�1
� (�)). But at that case, as one can easily see,
M(�) = X�1
� (�) ~M (�)X��1
��
(�) is a c.o. for (0.1). Theorem 1.2 is proved.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 171
V.I. Khrabustovsky
Note that in Th. 1.2 in the case I = R1 it is impossible in general to get rid
of (1.3) with F = N?, as one can see from
Example 1.2. Consider the block-diagonal equation (0.1) with
Q(t) = diagfGkg1k=1; H�(t) = diagfHk
� (t)g1k=1; (1.102)
with
Gk =
�
0 i
�i 0
�
; Hk
� =
�
0 0
0 1
�
+ �
�
�2k 0
0 0
�
; 0 < �k ! 0: (1.103)
(1.10) is valid for any solution of the form (1.9) for equations (0.1), (1.102), (1.103)
on I = R1 if and only if in (1.9)
M(�) =
i
2
diag
(
1
�k
p
�
0
0 �k
p
�
!)1
k=1
(1.104)
is an unbounded operator-valued function (I�f(1:71) 2 DM(�)).
Note that the Ex. 1.2 demonstrates the possibility of existence of an operator-
valued function M(�) with the following properties. It is densely de�ned and
unbounded in H(= N?). However, (1.9) with this M(�) determines on a dense
in L2
w�
(I) (w� = wi) linear manifold a bounded in L2
w�
(I) analytic on � by [30,
p. 195] operator-valued function R� = R��� which satis�es (1.65).
Remark 1.2. If one writes down the c.o's M(�) produced in the proof of
Th. 1.2, Cases I, II, in the form (1.20), then the corresponding operator-valued
function P(�) is a projection, i.e., P2(�) = P(�).
Besides that, for those c.o.'s (1.66) is valid even if one replaces in (1.67) P
with I.
The proof of the �rst statement follows from
Lemma 1.9. Let P 2
n = Pn 2 B(H), KerPn = K does not depend on n,
P = w � limPn. Then P 2 = P , KerP = K.
P r o o f o f L e m m a 1.9. Let f 2 K. Then Pf = w � limPnf = 0,
hence K � KerP . Now assume h 2 H. Then since (I � Pn)h 2 K, one has
(I � P )h = w � lim(I � Pn)h 2 K due to [30, p. 177]. Therefore if for all h 2 H
P (I �P )h = 0, then one has P 2 = P and KerP = K since KerP = (I �P )H �
K. Lemma 1.9 is proved.
The second statement follows from the fact that Mn(�) (1.97) is a c.o. for the
equation (0.1), (1.92), (1.93) on the interval (a; �n + 1) where P = I. Thus for
Mn(�) (1.97) one has (1.66) for I = (a; �n) if P = I in (1.67). Therefore while
proving Lem. 1.6 (which in fact gives the desired c.o.) one may set P = Pn = I
in (1.90), (1.91), together with the subsequent argument. The remark is proved.
172 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On the Characteristic Operators and Projections and on the Solutions of Weyl...
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