On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems.III. Separated Boundary Conditions
For the systems as in the title, boundary-value problems with separated boundary conditions are considered. We prove that the characteristic operator of such problem admits a special expression in terms of the projection (characteristic projection). This allows one to introduce for the above systems...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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| Цитувати: | On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems.III. Separated Boundary Conditions / V.I. Khrabustovsky // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 449-473. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1066792016-10-03T03:02:15Z On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems.III. Separated Boundary Conditions Khrabustovsky, V.I. For the systems as in the title, boundary-value problems with separated boundary conditions are considered. We prove that the characteristic operator of such problem admits a special expression in terms of the projection (characteristic projection). This allows one to introduce for the above systems the analogues of theWeyl functions and solutions, to establish for them the Weyl type inequalities which turn out to be well known in a number of special cases. 2006 Article On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems.III. Separated Boundary Conditions / V.I. Khrabustovsky // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 449-473. — Бібліогр.: 5 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106679 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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For the systems as in the title, boundary-value problems with separated boundary conditions are considered. We prove that the characteristic operator of such problem admits a special expression in terms of the projection (characteristic projection). This allows one to introduce for the above systems the analogues of theWeyl functions and solutions, to establish for them the Weyl type inequalities which turn out to be well known in a number of special cases. |
| 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.III. Separated Boundary Conditions 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.III. Separated Boundary Conditions |
| title_short |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems.III. Separated Boundary Conditions |
| title_full |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems.III. Separated Boundary Conditions |
| title_fullStr |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems.III. Separated Boundary Conditions |
| title_full_unstemmed |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems.III. Separated Boundary Conditions |
| title_sort |
on the characteristic operators and projections and on the solutions of weyl type of dissipative and accumulative operator systems.iii. separated boundary conditions |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106679 |
| citation_txt |
On the Characteristic Operators and Projections and on the Solutions of Weyl Type of Dissipative and Accumulative Operator Systems.III. Separated Boundary Conditions / V.I. Khrabustovsky // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 449-473. — Бібліогр.: 5 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT khrabustovskyvi onthecharacteristicoperatorsandprojectionsandonthesolutionsofweyltypeofdissipativeandaccumulativeoperatorsystemsiiiseparatedboundaryconditions |
| first_indexed |
2025-07-07T18:51:23Z |
| last_indexed |
2025-07-07T18:51:23Z |
| _version_ |
1837015274094592000 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 4, pp. 449�473
On the Characteristic Operators and Projections
and on the Solutions of Weyl Type of Dissipative
and Accumulative Operator Systems.
III. Separated Boundary Conditions
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
For the systems as in the title, boundary-value problems with separated
boundary conditions are considered. We prove that the characteristic ope-
rator of such problem admits a special expression in terms of the projection
(characteristic projection). This allows one to introduce for the above sys-
tems the analogues of the Weyl functions and solutions, to establish for them
the Weyl type inequalities which turn out to be well known in a number of
special cases.
Key words: operator di�erential equation, characteristic operator, cha-
racteristic projection, separated boundary conditions, Weyl function, Weyl
type solution, maximal semi-de�nite subspace.
Mathematics Subject Classi�cation 2000: 34B07, 34B20, 34G10, 47A06.
This work constitutes Part III of [36]. Notation, de�nitions, numeration of the
sections, statements, formulas etc., as well as the list of references extend those
of [36].
3. Separated Condition (1.10). Characteristic Projection
De�nition 3.1. Let M(�) be a c.o. for equation (0.1) on I. We say that
condition (1.10) is separated with a nonreal � = �0 if for any H-valued function
f(t) 2 L2
w�0
(I) with a compact support solution x�0(t) (1.9), for � = �0 of (0.1)
satis�es
lim
t#a
=�0U [x�0(t)] � 0; lim
t"b
=�0U [x�0(t)] � 0:? (3.1)
?Every limit in (3.1) exists in view of (1.11).
c
V.I. Khrabustovsky, 2006
V.I. Khrabustovsky
Note that ifM(�) is a c.o. and in one of inequalities (3.1) there is an equality,
then another one holds automatically due to (1.10).
The following statement admits a proof similar to that of n020 of Th. 1.1.
Remark 3.1. The validity of (3.1) with a nonreal � = �0 for an operator
function M(�) 2 B(H) of the form (1.20) and any H-valued vector function
f(t) 2 L2
w�0
(I) with compact support is equivalent to
8t 2 �I � =�0�
�
�0
(t) � 0 (3.2)
with ��
�
(t) being as in (1.67).
Theorem 3.1. Let P = I, M(�) (1.20) be a c.o. of (0.1), =�0 6= 0. Then
condition (1.10) corresponding to M(�) is separated with � = �0 if and only if
P2(�0) = P(�0): (3.3)
P r o o f. Suppose that condition (1.10) is separated with � = �0. Take in (3.2)
t = c. Then one has: =�0P
�(�0)GP(�0) � 0, =�0(I �P
�(�0))G(I �P(�0)) � 0,
which implies (3.3) in view of (1.69) and Ths. 2.4, 2.7.
Conversely, suppose (3.3) holds. By no2o of Th. 1.1 one has
8[�; �] � �I : =�0(P
�(�0)X
�
�0
(�)Q(�)X�0
(�)P(�0)
�(I �P�(�0))X
�
�0
(�)Q(�)X�0
(�)(I �P(�0))) � 0: (3.4)
Multiply (3.4) by P�(�0) from the left and by P(�0) from the right to deduce
that 8� 2 �I =�0�
+
�0
(�) � 0. In a similar way one can establish that 8� 2 �I
=�0�
�
�0
(�) � 0, so the theorem is proved in view of Remark 3.1.
As a consequence of (1.68), formula (9) of [1] and Ths. 3.1, 2.4, 2.7, formula
(1.69) we have
Corollary 3.1. Let P = I,M(�) (1.20) be a c.o. of (0.1) on I. Then in order
to claim that condition (1.10) is separated with a nonreal � = �0, it is necessary
to have simultaneously the two inequalities
(I �P�(�0))��0
(�; c)(I �P�(�0)) �
1
2=�0
(I �P�(�0))G(I �P(�0));
P�(�0)��0
(c; �)P(�0) � �
1
2=�0
P�(�0)GP(�0);
(3.5)
for all �nite � � c � �, [�; �[� �I, and it is su�cient to have simultaneously the
two inequalities (3.5) with � = c = �.
450 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Characteristic Operators and Projections and on the Solutions of Weyl...
Remark 3.2. If M(�);P(�) 2 B(H) are related by (1.20), then
M(�) =M�(��)() P(�) = G�1(I �P�(��))G
() (I �P�(��)G(I �P(�)) = P�(��)GP(�) (3.6)
and hence �
M(�) =M�(��)
�
^
�
(P2(�) = P(�)
�
() (I �P�(��)G(I �P(�)) = P�(��)GP(�) = 0: (3.7)
The following Remark 3.3. establishes a relationship between a c.o. with
the separated condition (1.10) and the boundary-value problems with separated
boundary conditions which depend on the spectral parameter.
Remark 3.3. Suppose the interval I = (a; b) is �nite and condition (1.3)
holds with F = H. Then:
10. If operator functions M�;N� from n010 of Remark 1.1 are such that
M�
��
Q(a)M� = N �
��
Q(b)N�, =�M
�
�
Q(a)M� � 0, =�N �
�
Q(b)N� � 0, =� 6= 0
(i.e., boundary condition (1.14), (1.13) is separated), then a solution of boundary-
value problem (0.1), (1.14), (1.13) for any H-valued f(t) 2 L2
w�
(I) is given by
x�(t) (1.9), where M(�) is a c.o. of (0.1) on I for which condition (1.10) is
separated. Hence M(�) admits representation (1.20), with P(�) being a projection
which, as one can easily see, is just
P(�) = �X�1
�
(b)N�
�
X�1
�
(a)M(�) �X�1
�
(b)N (�)
��1
(3.8)
where (:::)�1 2 B(H). In this setting, boundary condition (1.14), (1.13) is sepa-
rated ,
�
M�
��
Q(a)M� = 0
�
_
�
N�
��
Q(b)N� = 0
�
, =� 6= 0.
20. If M(�) (1.20) is a c.o. of (0.1) on I in such a way that P2(�) = P(�),
then x�(t) (1.9) is a solution of some boundary-value problem from n010 of Re-
mark 1.1 with separated boundary condition (1.14, (1.13)).
P r o o f. All the claims of n010; except the last one, follow from n010 of
Remark 1.1. As for the last claim of n010, it follows from
M�
��
Q(a)M� = N �
��
Q(b)N�
= (X�1
��
(a)M�� �X�1
��
(b)N��)
�GP(�)(I �P(�))
�
X�1
�
(a)M� �X�1
�
(b)N�
�
;
which is a consequence of (1.1), (1.15), (1.20), (3.6).
The claim 20 follows by an argument of proof of n020 of Remark 1.1, Th. 3.1,
Remark 3.1. Remark 3.3 is proved.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 451
V.I. Khrabustovsky
Theorem 3.2. Let P = I;M(�) be a c.o. of (0.1) on I: Then one has:
1. If condition (1.10) is separated with some nonreal �, then it is separated
with ��.
2. Suppose I is �nite and there exist �0, �0 such that =�0=�0 > 0 and
the following is true: 1) condition (1.10) with � = �0 becomes an equality; 2)
condition (1.10) with � = �0 is separated. Then condition (1.10) is separated
with any nonreal �.
3. If in (0.1) one has H�(t) = H0(t) + �H(t);H0(t) = H�
0 (t) and (1.3) holds
with F = H, then n020 is valid without assuming �niteness of I.
P r o o f. 10 follows from Th. 3.1 and Remark 3.2.
20 follows from Remarks 1.1, 3.3 and Th. 2.8.
30. Without assumption P = I, consider in L2
H
(I) � L2
H
(I) a linear ma-
nifold L00 =
n
y(t) � g(t)
��� y(t) L
2
H
(I)
= x(t); x(t) 2 ACloc is a vector function
with compact support; g(t)
L
2
H
(I)
= f(t); f(t) is an H-valued vector function and
l[x] = H(t)f(t); with l[x(t)] = i
2
((Q(t)x(t))0 +Q(t)x0(t)) �H0(t)x(t)
o
, which is
symmetric [37, p. 75] by the Lagrange formula.
In what follows we replace generic elements of L00 by the elements of the form
x(t)� f(t). Introduce the notation L0 = �L00.
In the case whenM(�) is a c.o., we keep the notation R(�) for an extension by
a continuity onto L2
H
(I) of the operator R�(1.9) originally de�ned on H-valued
vector functions f(t) 2 L2
H
(I) with compact support, which is bounded by n01
of Th. 1.1.
We need the following two Lemmas; the �rst one does not require P = I. In
the statements and in the proofs of Lems. 3.1, 3.2 and Remark 3.3 the notation
of operator and its graph coincide.
Lemma 3.1. R(�) is a generalized resolvent of the subspace L0 (in the sense
of formula (6.25) of [37]).
P r o o f. Note that R�(�) = R(��) since M�(�) = M(��), that R(�) satis�es
(1.65) with R� = R(�) and that R(�) depends analytically on � by [30, p. 195].
These observations imply, in view of Th. 6.8 of [37], that in our case to �nish
proving of the Lem. 3.1. it remains to verify that
R(�)(L0 � �) � I; (3.9)
with I being the graph of an identity operator in L2
H
(I).
If L00 = fx(t) � f(t)g; then
R(�)(L00 � �) = x(t)� x�(t);
452 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Characteristic Operators and Projections and on the Solutions of Weyl...
with x�(t) = R�(f(t)� �x(t)). Introduce the notation z�(t) = x�(t)� x(t).
If suppx(t) � [�; �] � �I, then by (1.10) one has
=�((Q(�)z�(�); z�(�))� (Q(�)z�(�); z�(�))) � 0: (3.10)
On the other hand,
(Q(�)z�(�); z�(�)) � (Q(�)z�(�); z�(�)) = 2=�
�Z
�
(H(t)z�(t); z�(t))dt; (3.11)
since
l[z�(t)] = �H(t)z�(t):
It follows from (3.10), (3.11) that z�
L
2
H
(I)
= 0, hence
R(�)(L00 � �) � I: (3.12)
Suppose y(t)� g(t) 2 L0 and L
0
0 3 yn(t)� gn(t)! y(t)� g(t). Then by (3.12)
one has
R(�)
�
y(t)� [g(t)� �y(t)]
�
= limR(�)
�
yn(t)� [gn(t)� �yn(t)]
�
= lim(yn(t)� yn(t)) = y(t)� y(t);
so that (3.9), and hence Lem. 3.1, are proved.
Lemma 3.2. Let H be an arbitrary Hilbert space, R(�) the generalized resol-
vent of a symmetric subspace S � H2, and
9�0; =�0 6= 0; 8f 2 H : kR(�0)fk
2 =
=(R(�0)f; f)
=�0
: (3.13)
Then:
10. (3.13) is valid for all � with =�=�0 > 0.
20.
R(�) =
(
( ~S � �)�1; =�=�0 > 0;
( ~S� � �)�1; =�=�0 < 0;
(3.14)
with ~S being a maximal symmetric extension of S.
P r o o f. 10. It is known from [37, p. 95] that
R(�) = (T (�)� �)�1; (3.15)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 453
V.I. Khrabustovsky
with the linear subspace T (�) � H2 being such that
=T (�) � 0(max); =� > 0; (3.16)
T (��) = (T (�))�: (3.17)
Its Cayley transform C�(T (�)) with �xed �, =�=� > 0, is a contraction in H
which depends analytically on �.
Since with =�=� 6= 0, f 2 H one has
k(I + (�� ��)R(�))fk2 = kfk2 + 4(=�)2
�
kR(�)fk2 �
=(R(�)f; f)
=�
�
;
C�0
(T (�0)) is an isometry in H in view of formula (4.17) from [37]. Thus C�0
(T�)
= C�0
(T�0) with =�=�0 > 0 by [35, p. 210], hence 10 is proved in view of formula
(4.17) from [37].
20. Use 10 to deduce from Th. 6.7, the formulas (6.20)�(6.24) from [37] and
(3.16) that T (�) = T (�0)
def
= ~S with =�=�0 > 0, where ~S is the maximal sym-
metric extension of S by Th. 6.2 from [37]. Thus 20 is proved in view of (3.15),
(3.17). Lemma 3.2 is proved.
Turn back to the proof of Th. 3.2. It is clear from the proof of n06 of Th. 1.1
that if (1.3) holds with F = H, then the integral
x�(t) =
bZ
a
K(t; s; �)H(t)f(t)dt; (3.18)
with K(t; s; �) being as in (1.85), converges and x�(t) 2 L2
H
(I) even in the case
when an H-valued f(t) 2 L2
H
(I) does not have a compact support. Prove that in
this case inequalities (3.1) hold for x�0(t) (3.18), if they hold for x�0(t)(3.18), (� =
�0) with f(t) having a compact support. Let fn(t) = �n(t)f(t) with �n(t) being
characteristic functions of the intervals (�n; �n) " (a; b). In view of convergence
of (3.18) and (1.70), it is possible to choose for any " > 0 such N" that for all
n � N" one has
kx�0(c)� xn
�0
(c)k < "; (3.19)
with xn
�0
(t) being given by (3.18) in which � = �0, f(t) = fn(t),
kx�0(t)� xn�0(t)kL2
H
(I) < "; (3.20)
8� 2 (a; c)
1
2
��(U [x�0(c)]� U [x�0(�)]) � (U [xn�0(c)] � U [xn�0(�)])
��
� j=�0j
�
kx�0(t)k
2
L
2
H
(�;c)
� kxn�0(t)k
2
L
2
H
(�;c)
�
+
���=((x�0(t); f(t))L2
H
(�;c) � (xn�0(t); fn(t))L2
H
(�;c))
��� < �: (3.21)
454 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Characteristic Operators and Projections and on the Solutions of Weyl...
Thus in view of (3.19)�(3.21)
U [x�0(�)] � U [xn
�0
(�)]! 0
uniformly in � 2 (a; c). Therefore, for the following limit which exists due to
(1.70), one has
lim
t#a
=�0U [x�0(t)] = lim
�n#a
=�0U [x�0(�n)] = lim
�n#a
=�0U [x
n
�0
(�n)] � 0:
The second inequality in (3.1) for x�0(t) admits a similar veri�cation. After that
n03 follows from the Hilbert identity for R(�), which is valid if =�=�0 > 0 in
view of Lem. 2.4 from [37] and (1.70), Lems. 3.1, 3.2. The Theorem 3.2 is proved.
Note that assumption 1) in n02 of Th. 2.2 could not be omitted in general, as
it follows from Remarks 1.1, 2.6.
We are about to expand Lem. 3.1 in the case when � is involved into H�(t)
nonlinearly as follows:
H�(t) = �H(�) +H1
�
(t); (3.22)
with H1
�
(t) satisfying the same conditions as H�(t), H(t) � 0.
Let M(�) be a c.o. of (0.1), (3.22). Then, if an H-valued f(t) 2 L2
H
(I) has
compact support and suppf(t) � [�; �] � �I, one has in view of n02 of Th. 1.1
that x�(t) (3.18) with =� 6= 0 satis�es the inequality
=�(U [x�(�)]� U [x�(�)]) � 0; (3.23)
since
R
b
a
X�
��
(t)H(t)f(t)dt 2 N? by (3.22).
Denote by <�f = x�(t) (3.18), with an H-valued f(t) 2 L2
H
(I) having a com-
pact support. Using (3.23), (3.22), one can prove, just as in the case of n01 of
Th. 1.1, that
k<�f)kL2
H
(I) �
=(<�f; f)L2
H
(I)
=�
; =� 6= 0: (3.24)
Denote by <(�) the extension by a continuity onto L2
H
(I) of <�f which is
bounded by (3.24). Note that <�(�) = <(��) since M�(�) = M(��), that <(�)
satis�es (3.24) with <� = <(�) and that <(�) depends analytically on � by [30,
p. 195]. Therefore, in view of Ths. 4.5, 3.2 from [37], we come to the following
Remark 3.4. <(�) = (T (�)��)�1, where the linear subspace T (�) � L2
H
(I)�
L2
H
(I) satis�es (3.16), (3.17), and its Cayley transform C�(T�); (see [37]) under
�xed � and =�=� > 0; is a contraction in L2
H
(I) which depends analytically on �.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 455
V.I. Khrabustovsky
De�nition 3.2. If an operator function M(�) 2 B(H) of the form (1.20)
is a c.o. of (0.1) on I in such a way that P(�) = P2(�), then P(�) is called
a characteristic projection (c.p.) of (0.1) on I (or merely a c.p.).
Theorem 3.3. A c.p. of (0.1) on I exists if one of the ends of I is �nite or
if for some �0 2 A \ R
1 the norm kX�
�0
(t)w�0
(t)X�0
(t)k is summable at one of
the ends of I. Also, a c. p. exists if (1.3) holds with F = H.
If Q(t) is de�nite, then a c.p. of (0.1) exists without any additional conditions.
P r o o f. By Lemma 1.2 and the proof of Th. 1.2, it is su�cient to prove
Th. 3.3 for (0.1) with an inde�nite constant Q(t) = G.
If one of the ends of I is �nite or the norm kX�
�0
(t)w�0
(t)X�0
(t)k is summable
at one of its ends, then, in view of Remark 1.2, the Th. 3.3 has already been
proved while proving Th. 1.2.
Assume I = R1 and (1.3) holds with F = H.
First, suppose that c = �. Consider the projection P+(�) associated via (1.20)
to the c.o. of (0.1) on (c;1), which is constructed within the scheme of the proof
of Th. 2.1, Case I (see Remark 1.2).
It follows from Th. 3.1, Cor. 3.1 and Lem. 1.1 that
9Æ(�) > 08f 2 H :
1
2=�
(P�+(�)GP+(�)f; f) � �Æ(�)kP+(�)fk
2 (=� 6= 0)
and by Remark 1.2 and Lem. 1.9 one has
9c+ > 0 8f 2 H : (Sgn=�)(I �P�+(�))G((I �P+(�))f)
� c+k(I �P+(�))fk
2 (=� 6= 0):
On the other hand, replace �(�) by I��(�) and then use the scheme described
in the proof of Case I, Th. 2.1 to produce a c.o. on (�1; c) for (0.1). Consider
the projection P�(�) associated with this c.o. by (1.20). By Remark 1.2 and
Lem. 1.9 for =� 6= 0
9c� > 0 8f 2 H : (Sqn=�)(P��(�)GP�(�)f; f) � �c�kP�(�)fk
2;
=�(I �P��(�))G(I �P�(�)) � 0:
Therefore by Lem. 2.4, Th. 2.4, and [25, p. 76] one has
H = (I �P�(�))H
:
+ P+(�)H:
Denote by P(�) the projection onto P+(�)H parallel to (I �P�(�))H.
By Lemma 1.3 and [1] (see also Cor. 2.3) one has
P(�) = P+(�)(P+(�) + I �P�(�))
�1;
456 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Characteristic Operators and Projections and on the Solutions of Weyl...
with (:::)�1 2 B(H). It is easy to see that P(�) is a desired c.p.
If c 6= �, produce in a similar way a c.p. P(�) for the case when
the Cauchi operator of (0.1) is normalized by I at �. Then the desired c.p. is
just X�1
�
(�)P(�)X�(�), so the Th. 3.3 is proved.
Note that with P 6= I (3.1) does not imply (3.3) and (3.3) does not imply
(3.1) even in the �nite dimensional case, as one can see from
Example 3.1. Let I = (0; 1), c = 0, in (0.1):
Q(t) =
�
0 i
�i 0
�
; H�(t) =
�
0 0
0 i=4
�
; =� > 0:
Then:
I. M(�) (1.20) is a c.o. with
P(�) =
�
0 1
0 2
�
6= P2(�); =� > 0:
However condition (1.10) is separated with =� > 0.
II. M(�) (1.20) is a c.o. with
P(�) =
�
i 1
1 + i 1� i
�
= P2(�); =� > 0:
However condition (1.10) is not separated with =� > 0.
With P 6= I one has the following analogue of Th. 3.1.
Theorem 3.4. Suppose P 6= I and 9�0 2 CnR
1,
2 �I:
N =
\
��
Ker��0
(�;
) or N =
\
��
Ker��0
(
; �):
LetM(�) be a c.o. of (0.1). If condition (1.10) associated toM(�) is separated
with nonreal � = �0, then
9M0 2 B(H) : PM0P = 0; M(�0) +M0 =
�
P �
1
2
I
�
(iG)�1:
Here the operator P is extendable from (GN)? to, possibly unbounded, den-
sely de�ned in H; idempotent which, in the case when either dimP�H < 1 or
dimP+H <1 with P� being as in (2.1), (2.2), is a bounded projection in H.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 457
V.I. Khrabustovsky
4. The Weyl Type Functions and Solutions
In this section a < c < b, unless a di�erent assumption is stated. Condition
(1.3) with F = H is valid for I = I+ = (a; c) if I+ 6= ; and I = I� = (c; b) if
I� 6= ;.
The following theorem describes a relationship between the c.p. on I and
on I�.
Theorem 4.1. 10. Let P(�) be a c.p. of (0.1) on I, P� = P�(�) 2 B(H)
be some operator-valued functions that depend analytically on nonreal � and such
that
�=�P ��GP� � 0; �=�(I � P ��)G(I � P�) � 0; =� > 0 or =� < 0; (4.1)
P�(�) = G�1(I � P ��(
��))G; =� 6= 0:? (4.2)
Then
H = P+(�)H
:
+ P(�)H = P�(�)H
:
+ (I �P(�))H: (4.3)
Denote also by P+(�) and P�(�) the projections onto P(�)H and P�(�)H,
respectively, parallel to P+(�)H and (I �P(�))H, respectively.
Then P�(�) are c.p. of (0.1) on I = I� in such a way that
P+(�) = P(�)(P(�) + P+(�))
�1;P�(�) = P�(�)(P�(�) + I �P(�))�1; (4.4)
with
(P(�) + P+(�))
�1; (P�(�) + I �P(�))�1 2 B(H): (4.5)
20. Let P�(�) be a pair of c.p. of (0.1) with
Q(t) = Q�(t); Q�(c) = G; H�(t) = H�
�
(t); t 2 I�; (4.6)
on I = I�, then H = P+(�)H
:
+ (I �P�(�))H.
Suppose P(�) projects onto P+(�)H parallel to (I �P�(�))H. Then P(�) is
a c.p. of (0.1),(4.6) on I = (a; b) in such a way that
P(�) = P+(�)S�(�)(P+(�)S�(�) + (I �P�(�))S+(�))
�1; (4.7)
where S+(�) and S�(�) are the Riesz projections for the operator (sgn=�)G that
correspond to positive and negative parts of its spectrum, respectively; (P+(�)S�(�)
+(I �P�(�))S+(�))
�1 2 B(H).
If the c.p. P�(�) is generated by the c.p. P(�) according to (4.4) in n010 of
the theorem, then n020 results exactly this P(�).
?(4.1) implies by Lem. 2.4, Ths. 2.4, 2.7, that P 2
�(�) = P�(�) for =� > 0 or =� < 0, so for
all nonreal � by (4.2).
458 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Characteristic Operators and Projections and on the Solutions of Weyl...
P r o o f. In view of (4.1), (4.2), Th. 2.4, Lem. 2.4, (3.6), (3.7), [25,
p. 73], P�(�)H and (I � P�(�))H are respectively maximal �=�G-nonnegative
and maximal �=G-nonpositive subspaces for nonreal �. In view of Corollary 3.1,
condition (3.1) with F = H for I = I� and [25, p. 71] (or Th. 2.4, (1.69) (or
Lem. 2.4)), P(�)H and (I �P(�))H are respectively maximal uniformly �=�G-
positive and maximal uniformly �=�G-positive subspaces for nonreal �. Hence
we have (4.3) by [25, p. 76] and (4.5) by Lem. 1.3. Thus we have (4.4) by [1] (or
Cor. 2.3) and hence P�(�) depends analytically on nonreal �. Thus P�(�) are
c.p. of (0.1) on I� since P(�) is a c.p. and by (3.6), (3.7), (4.2).
20 is proved similarly to 10. The Theorem is proved.
The following remark allows, in particular, to transform a c.p. so that the
corresponding to it boundary condition at one end of interval is not changed, but
boundary condition at another end coincides with any given. This Remark is
proved in the same way as Th. 4.1.
Remark 4.1. 10. a) Let ~P(�) be a c. p. of (0.1) on I. Then, if one sets
P+(�) = I � ~P(�), P�(�) = ~P(�) in 10 of Th. 4.1, then P�(�) (4.4) becomes
a c.p. of (0.1) not only on I�, but on I as well. b) Let ~P�(�) be a c.p. of
(0.1),(4.6) on I = I�. Then, if one sets P+(�) = I � ~P�(�), P�(�) = ~P+(�) in
10 of Th. 4.1, then (4.4) becomes a c.p. of (0.1) not only on I�, but also on I as
well with
Q(t) =
(
Q(t); t 2 I+
Q�(t); t 2 I�
; H�(t) =
(
H�(t); t 2 I+
H�
�
(t); t 2 I�
in the case of P+(�) and with
Q(t) =
(
Q+(t); t 2 I+
Q(t); t 2 I�
; H�(t) =
(
H+
�
(t); t 2 I+
H�(t); t 2 I�
in the case of P�(�).
20. If one replaces in (4.7) I � P�(�)(P+(�)) by P+(�)(P�(�)) as in 10 of
Th. 4.1, then one still has in (4.7) (:::)�1 2 B(H), however P(�) (4.7) in general
is no longer a c.p. of (0.1) on I, but a c.p. on I+(I�), that projects onto
P+(�)(P�(�)) parallel to P+(�)H((I �P�(�))H).?
We are about to demonstrate a procedure of producing the operator Weyl
type functions and solutions of (0.1) that uses projections from Ths. 3.1, 4.1.
In view of (1.22), (2.1), (2.2) it is easy to see that
9�(t) 2 B(H) : ��1(t) 2 B(H); �(t) 2 ACloc; ��(t)Q(t)�(t) = P+ � P�;
(4.8)
?The transformations of c.p. on I� such that they don't change boundary condition at the
point c construct in the similar way.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 459
V.I. Khrabustovsky
with P� being complementary orthogonal projections that do not depend on t.
Theorem 4.2. Let P(�) be a c.p. of (0.1) on (a; b). Then there exist unique
strict contractions K�(�) = K�
�(
��) that depend analytically on nonreal � such
that
K�(�) 2 B(P�H; P�H);=� > 0;K�(�) 2 B(P�H; P�H);=� < 0; (4.9)
P(�) =
8>>>><
>>>>:
�(c)(P� +K+(�)P�)(I� �K�(�)K+(�))
�1(P� �K�(�)P+)�
�1(c);
=� > 0;
�(c)(P+ +K+(�)P+)(I+ �K�(�)K+(�))
�1(P+ �K�(�)P�)�
�1(c);
=� < 0;
(4.10)
(here I� are the identity operators in P�H), and for the operator solutions
�(t; �) =
(
X�(t)�(c)(P� +K�(�)P�); =� > 0;
X�(t)�(c)(P� +K�(�)P�); =� < 0;
(4.11)
of the homogeneous equation (0.1), (4.8) one has
Z
J�
�
�(t; �)w�(t) �(t; �)dt �
(
1
2=�
P�(I� �K�
�(�)K�(�))P�; =� > 0;
�1
2=�
P�(I� �K�
�(�)K�(�))P�; =� < 0;
(4.12)
with J� being such �nite intervals that J� � (a; c), J+ � (c; b).
Conversely, suppose that for the operator functions K�(�) = K�
�(
��) that
depend analytically on nonreal � the relations (4.9), (4.11), (4.12) hold. Then
K�(�) are strict contractions and P(�) (4.10) is a c.p. of (0.1) (4.8) on (a; b).
P r o o f. Let P(�) be a c.p. of (0.1), (4.8) on (a; b). Then, in view of Cor. 3.1,
condition (3.1) with F = H for I = I� and [25, p. 71] (or (1.69) (or Lem. 2.4),
Th. 2.4), the subspaces
H�(�) = (I �P(�))H; H+(�) = P(�)H (4.13)
are respectively maximal uniformly �=�G-positive and maximal uniformly
�=�G-positive for nonreal �. Therefore [24, p. 100], [25, Ch. I, � 8] there exist
unique strict contractions K�(�) (4.9) such that
H�(�) =
(
�(c)fP�f �K�(�)P�f jf 2 Hg; =� > 0;
�(c)fP�f �K�(�)P�f jf 2 Hg; =� < 0:
(4.14)
460 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Characteristic Operators and Projections and on the Solutions of Weyl...
Thus (4.10) is valid for P(�). Substitute (4.10), (4.11) into (3.5) (with �0
being replaced by �) to get (4.12).
Prove that K�(�) = K�
�(
��) and depends on a nonreal � analytically. Intro-
duce the notation Hc(�) = Pc(�)H, with
Pc(�) =
(
�(c)P+�
�1(c); =� > 0
�(c)P��
�1(c); =� < 0:
(4.15)
By [25, p. 76] one has
H = Hc(�)
:
+H+(�):
Denote by P+(�) the projection onto H+(�) parallel to Hc.
By (3.6)
P+(�) = G�1(I �P�+(
��)))G; (4.16)
and, obviously
P+(�) =
(
�(c)(P� +K+(�)P�)�
�1(c); =� > 0;
�(c)(P+ +K+(�)P+)�
�1(c); =� < 0:
(4.17)
Compare (4.8), (4.16), (4.17) to observe thatK+(�) = K�
+(
��). By Th. 4.1 the ope-
rator function P+(�) is a c.p. on (c; b), hence it depends analytically on a nonreal
�. Thus K+(�) depends analytically on nonreal � in view of (4.17).
The same properties for K�(�) can be proved in a similar way.
Conversely, suppose that (4.9), (4.11), (4.12) are valid for the operator func-
tions K�(�) = K�
�(
��) that depend analytically on a nonreal �. In view of condi-
tion (1.3) with F = H for I = I�, one can easily deduce from (4.12) that
9Æ1 = Æ1(�) > 0 :
(
8f 2 P�H : ((I� �K�
�K�)f; f) � Æ1kfk
2; =� > 0;
8f 2 P�H : ((I� �K�
�K�)f; f) � Æ1kfk
2; =� < 0:
It follows that (4.10) determines an operator P(�) 2 B(H) which depends
analytically on a nonreal �. Thus (4.12), (4.11) imply (3.5), (�0 = �) and so
(1.68) (by (9) of [1]) with P(�) as in (4.10). Consider the operator P+(�) (4.17)
together with its analogue P�(�) for (b; c), which are obviously projections. Since
P�(�) satis�es relations similarly to (4.16), it follows from (3.6), (3.7) that such
a relation is valid for P(�). Hence P(�) is a c.p. in view of Th. 1.1, Cor. 3.1 and
Th. 3.1. The Theorem 4.2 is proved.
We call the operator functions K�(�) = K�
�(
��), K+(�) = K�
+(
��) (4.9), which
depend analytically on a nonreal �, and which satisfy (4.12), (4.11), the Weyl
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 461
V.I. Khrabustovsky
functions of (0.1), (4.8) on (a; c) and on (c; b) respectively (by a similarity to
[13]�[17])?. We call the corresponding solutions �(t; �) = �(t; �)
��
P�H and
+(t; �) = +(t; �)
��
P�H (�=� > 0) the Weyl solutions of (0.1), (4.8) on (a; c)
and on (c; b) respectively.
Theorem 4.3. Suppose that the interval (a; b) is �nite and the operator func-
tions K�(�) and K+(�) are the Weyl functions of (0.1), (4.8) on (a; c) and on
(c; b), respectively. Then there exist unique contractions U�(�) = U��(
��) that
depend analytically on a nonreal � and such that:
U�(�) 2 B(P�H; P�H); =� > 0; U�(�) 2 B(P�H; P�H); =� < 0; (4.18)
P�K�(�)P+ = P��
�1(c)(I ���(�))�(c)P+ (=� > 0);
P+K�(�)P� = P+�
�1(c)(I ���(�))�(c)P� (=� < 0);
(4.19)
P+K+(�)P� = P+�
�1(c)�+(�)�(c)P� (=� > 0);
P�K+(�)P+ = P��
�1(c)�+(�)�(c)P+ (=� < 0);
(4.20)
with
I ���(�) =
=
8>>>><
>>>>:
X�1
�
(a)�(a)(P+ + U�(�)P+)(X
�1
�
(a)�(a)(P+ + U�(�)P+)� P��(c))
�1;
=� > 0;
X�1
�
(a)�(a)(P� + U�(�)P�)(X
�1
�
(a)�(a)(P� + U�(�)P�)� P+�(c))
�1;
=� < 0;
(4.21)
�+(�) =
=
8>>>><
>>>>:
X�1
�
(b)�(b)(P� + U+(�)P�)(X
�1
�
(b)�(b)(P� + U+(�)P�)� P+�(c))
�1;
=� > 0
X�1
�
(b)�(b)(P+ + U+(�)P+)(X
�1
�
(b)�(b)(P+ + U+(�)P+)� P��(c))
�1;
=� < 0:
:
(4.22)
(In (4.22), (4.21) one has (:::)�1 2 B(H)). The operators ��(�) and �+(�) are
c.p. of (0.1) on (a; c) and on (c; b), respectively, and x�(t) (1.9), (1.20), (4.10)
?Note that in view of no5o of Th. 1.1 the validity of inequalities (4.11), (4.12) for arbitrary
K�(�) (4.9) in one of the complex half-planes implies validity of their analogs in another half-
plane if one sets up K�(��) = K
�
�(�).
462 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Characteristic Operators and Projections and on the Solutions of Weyl...
is a solution of the boundary-value problem (0.1), (1.14) with
M� =
(
�(a)(P+ + U�(�)P+); =� > 0;
�(a)(P� + U�(�)P�); =� < 0;
(4.23)
N� =
(
�(b)(P� + U+(�)P�); =� > 0;
�(b)(P+ + U+(�)P+); =� < 0:
Conversely, suppose that (4.18) holds for the contractions U�(�) = U��(
��)
depending analytically on a nonreal �. Then the operators (:::)�1 2 B(H) in
(4.22), (4.21) and operators (4.19), (4.20), associated to (4.22), (4.21), are the
Weyl functions of (0.1) on (a; c) and (c; b), respectively.
P r o o f. Let K+(�) be a Weyl function of (0.1), (4.8) on (c; b). Then, as
one can observe from the proof of Th. 4.2, P+(�) (4.17) is a c.p. of (0.1), (4.8)
on (c; b). Therefore the subspace X�(b)P+(�)H is maximal �=�Q(b) nonpositive
by Cor. 3.1, [25, p. 71] (or (1.69) (or Lem. 2.4), Th. 2.4), Lem. 2.6. Hence [24,
p. 100], [25, Ch. I, � 8] there exists such a unique contraction U+(�) (4.18) that
X�(b)P+(�)H =
(
�(b)(P� + U+(�)P�)H; =� > 0;
�(b)(P+ + U+(�)P+)H; =� < 0:
(4.24)
Since by (1.1) and Remark 3.2
P�+(
��)X�
��
(b)Q(b)X�(b)P+(�) = 0;
and hence one has U(�) = U�(��).
Consider the boundary-value problem (0.1), (4.8), (1.14) on (a; b) = (c; b) with
M� = Pc(�)�(c) with Pc(�) (4.15), N (�) (4.23).
It satis�es all the assumptions of Remarks 1.1, 3.3 by Cor. 2.1 (except analy-
ticity for N (�) so far). Thus projection (3.8) associated with the above problem
is just �+(�) (4.22). On the other hand,
�+(�)H =
(
X�1
�
�(b)(P� + U+(�)P�)H; =� > 0
X�1
�
�(b)(P+ + U+(�)P+)H; =� < 0
= P+(�)H;
(I ��+(�))H = (I �P+(�))H
by (4.24), (4.17). Hence P+(�) = �+(�), which, together with (4.17), implies
(4.20).
Analyticity for U+(�) = U�+(
��) is deducible from the fact that with =� > 0,
�(c)(P� + U+(�)P�)�
�1(c) = Y�(b)P+(�)(Y�(b)P+(�) + Pc(�))
�1;
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 463
V.I. Khrabustovsky
where (:::)�1 2 B(H), Y�(t) see Lem. 1.2. The claims related to K�(�), U�(�)
can be proved in a similar way.
Use (4.24) and its analogue for the endpoint a to deduce that x�(t) (1.9),
(1.20), (4.10) is a solution of the boundary-value problem (0.1), (1.14), (4.23).
Conversely, suppose that a contraction U+(�) = U�+(
��) which depends ana-
lytically on a nonreal �, satis�es (4.18). By Remarks 1.1, 3.3, a c.p. of problem
(0.1), (4.8), (1.14) on (a; b) = (c; b) withM� = Pc(�)�(c);N� (4.23) is just �+(�)
(4.22). Hence (see the proof of Th. 4.2) there exists the Weyl function K+(�) of
(0.1), (4.8) on (c; b) such that �+(�) = P+(�) (4.17) ) (4.20), (4.22) for K+(�).
The statements related to U�(�) can be proved in a similar way. The Theorem 4.3
is proved.
Theorem 4.4. Let P(�) be a c.p. of (0.1), (4.8) on (a; b) = (c; b). Then there
exist the unique Weyl function K+(�) of this equation on (c; b) and the unique
contraction U�(�) = U��(
��) (4.18) which depends analytically on a nonreal �,
such that
P(�) =
8>>>><
>>>>:
�(c)(P� +K+(�)P�)(I� � U�(�)K+(�))
�1(P� � U�(�)P+)�
�1(c);
=� > 0;
�(c)(P+ +K+(�)P+)(I+ � U�(�)K+(�))
�1(P+ � U�(�)P�)�
�1(c);
=� < 0;
(4.25)
and for any H-valued vector function f(t) 2 L2
w�
(c; b) with compact support the
solution of (0.1), (4.8) x�(t) (1.9), (1.20), (4.25) satis�es at c the following boun-
dary condition:
9h = h(f; �) : y(c) =M�h; (4.26)
with M� as in (4.23).
Conversely, let K+(�) be an arbitrary Weyl function of (0.1), (4.8) on (c; b)
and U�(�) = U��(
��) (4.18)be an arbitrary contraction that depends analytically
on nonreal �. Then (4.25) is a c.p. of (0.1), (4.8) on (c; b).
P r o o f. Let P(�) be a c.p. of (0.1), (4.8) on (c; b). Thus by Th. 3.1
and Cor. 3.1, (3.5) is valid with a = c and �0 being replaced by any nonreal �.
Therefore the subspaces H�(�) (4.13) and H+(�) (4.13) are maximal and =�G-
nonnegative and uniformly negative, respectively. Hence [24, p. 100], [25, Ch. I,
� 8] there exist the unique contraction U�(�) (4.18) and the unique strict con-
traction K+(�) (4.9) such that (4.14) holds with K�(�) being replaced by U�(�).
Thus P(�) satis�es (4.25), whence (4.26). Substitute (4.25) into (3.5) (with �0
being replaced by �) to deduce that K+(�) satis�es (4.12). The proof of relations
464 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Characteristic Operators and Projections and on the Solutions of Weyl...
K+(�) = K�
+(�), U�(�) = U��(
��) goes through like that of the similar relations
for K�(�) in Th. 4.2.
Analyticity for K+(�) and U�(�) follows by Lem. 3.1 from
P+(�) = P(�)(P(�) + Pc(�))
�1; (4.27)
�(c)(P+ + U�(�)P+)�
�1(c) = (I �P(�))(I �P(�) + I � Pc(�))
�1; =� > 0;
respectively, with P+(�) as in (4.17), Pc(�) as in (4.15), and (:::)�1 2 B(H).
Conversely, let K+(�) be the Weyl function of (0.1), (4.8) on (c; b) and
U�(�) = U��(
��) (4.18) be a contraction that depends analytically on a nonreal �.
Then (4.25) determines the operator P(�) 2 B(H) which is a c.p. in view of proof
of Th. 4.2 and M�
��
GM� = 0. The Theorem 4.4 is proved.
Lemma 1.3 and the proof of Th. 4.2 imply
Remark 4.2. K+(�) (4.9) is a Weyl function of (0.1), (4.8) on (c; b) if and
only if P+(�) (4.17) is a c.p. of this equation on (c; b). This c.p., hence also
K+(�), can be derived from the c.p. P(�) (4.25) using (4.27), (4.15).
Remark 4.3. Let P(�) be a c.p. of equation (0.1), (4.8) on (c; b) (hence
P(�) admits representation (4.25)), and with a nonreal � = �0 and any H-valued
vector functions f(t) 2 L2
w�0
(c; b) with the compact support for solutions x�0(t)
(1.9), (� = �0) of this equation corresponding to M(�0) (1.20), (4.25), (� = �0)
one has
lim
�"b
U [x�0(�)] = 0: (4.28)
Then inequality (4.12) for the solution +(t; �0) (4.11) becomes an equality with
� = �0 and J+ being replaced by (c; b)?.
Conversely, let K+(�) be the Weyl function of (0.1), (4.8), and suppose that
for some nonreal � = �0 and the associated solution +(t; �0) (4.11) of (0.1),
(4.8) on (c; b), inequality (4.12) becomes the equality with J+ being replaced by
(c; b)?. Let the contraction U�(�) satis�es (4.18), (� = �0). Then with � = �0
and any H-valued vector functions f(t) 2 L2
w�0
(c; b) with compact support, (4.28)
holds for solutions x�0(t)(1.9), (1.20), (4.25), (� = �0).
P r o o f. Assume for certainty =�0 > 0. It follows that one can use vector
functions of the form (1.75) with compact support to deduce from (4.28) that
s� lim
�"b
U [X�0
(�)P(�0)] = 0: (4.29)
?Where
bR
c
= s� lim
J+"(c;b)
R
J+
.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 465
V.I. Khrabustovsky
Therefore it is easy to conclude from (4.27) that
s� lim
�"b
U [X�0
(�)P+(�0)] = 0; (4.30)
with P+(�) being as in (4.17). On the other hand, by (1.77) and formula (9) from
[1] one has
P�+(�0)��0
(c; �)P+(�0) =
1
2=�0
(U [X�0
(�)P+(�0)]� U [X�0
(c)P+(�0)]) (4.31)
whence in view of (4.17),
�Z
c
�
+(t; �0)w�0
(t) +(t; �0)dt
=
1
2=�0
�
P�(I� �K�
+(�0)K+(�0))P� + U [X�0
(�)P+(�0)�(c)]
�
; (4.32)
so that the equality in (4.10) with � = �0 and J+ being replaced by (c; b), is
proved by a virtue of (4.30).
Conversely, suppose that the relation just proved is true. Then (4.32), (4.31)
imply (4.30), hence in view of (4.25) one has also (4.29), which implies (4.28).
The Remark 4.3 is proved.
As the consequence of Th. 4.3 proof, [35, p. 210], Remark 4.1, Lemmas 3.1,
3.2 we have
Remark 4.4. Let P(�) be a c.p. of equation (0.1), (4.8) on (c; b). Then
condition (4.28) implies the similar condition for � such that =�=�0 > 0, if
b <1 or if H�(t) = H0(t) + �H(t), H0(t) = H�
0 (t).
The statements, which are similar to Th. 4.4, Remarks 4.2�4.4 hold for the
interval (a; c).
Classes of equations (0.1) such that with dimH <1 (4.28) holds for any c.p.,
are described in [38, 39].
We illustrate below Ths. 4.2�4.4 in three basic cases. To simplify notation
assume that =� > 0.
I. Let H = H1 � H2, Q(t)
�
I1 0
0 �I2
�
, with Ij; j = 1; 2 being the identity
operators in Hj.
In this case one has
� = I; P+ =
�
I1 0
0 0
�
; P� =
�
0 0
0 I2
�
;
466 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Characteristic Operators and Projections and on the Solutions of Weyl...
and the Weyl functions K+(�) 2 B(H2;H1), K�(�) 2 B(H1;H2) (contractions
U�(�) act in a similar way). Therefore
P� +K+(�)P� =
�
0 K+(�)
0 I2
�
;
P� �K�(�)P+ =
�
0 0
�K�(�) I2
�
;
hence the projection P(�) (4.10) in Th. 4.2 is just
P(�) =
�
K+(�)
I2
�
(I2 �K�(�)K+(�))
�1(�K�(�); I2);
and the projection P(�) (4.25) in Th. 4.4 is given by
P(�) =
�
K+(�)
I2
�
(I2 � U�(�)K+(�))
�1(�U�(�); I2):
The Weyl solutions are given by
�(t; �) =
�
x1(t; �) + x2(t; �)K�(�)
x3(t; �) + x4(t; �)K�(�)
�
; +(t; �) =
�
x1(t; �)K+(�) + x2(t; �)
x3(t; �)K+(�) + x4(t; �)
�
with
x1(t; �) 2 B(H1); x2(t; �) 2 B(H2;H1);
x3(t; �) 2 B(H1;H2); x4(t; �) 2 B(H2);�
x1(t; �) x2(t; �)
x3(t; �) x4(t; �)
�
= X�(t): (4.33)
The inequalities (4.12) are equivalent toZ
J�
��(t; �)w�(t) �(t; �)dt �
1
2=�
(I� �K�
�(�)K�(�)); I+ = I1; I� = I2:
The operators (4.19), (4.20) from Th. 4.3 are given by
P�K�(�)P+ =
�
0 0
K�(�) 0
�
; P+K+(�)P� =
�
0 K+(�)
0 0
�
;
with the Weyl functions K�(�), by a virtue of (4.22), (4.21), being (in the case
dimH <1 cf. [8])
K�(�) = �(x�2(a;
��)� x�4(a;
��)U�(�))(x
�
1(b;
��)� x�3(b;
��)U�(�))
�1; (4.34)
K+(�) = �(x�1(b;
��)U+(�)� x�3(b;
��))(x�2(b;
��)U+(�)� x�4(b;
��))�1; (4.35)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 467
V.I. Khrabustovsky
(in (4.34) (:::)�1 2 B(H1); in (4.35) one has (:::)�1 2 B(H2)).
II. Let
H = H1 �H1 = H2
1; Q(t) =
�
0 iI1
�iI1 0
�
: (4.36)
In this case one has
� = I; P� =
1
2i
�
I1
�iI1
�
(iI1;�I1):
It is easy to see that the following representations are valid
P�K�(�)P� =
1
2i
�
�ik�(�)
k�(�)
��
iI1;�I1
�
;
with k�(�) being strict contractions in H1, so the projection P (4.10) in Th. 4.2
is given by
P(�) =
1
2i
�
I1 + ik+(�)
iI1 + k+(�)
�
(I1 � k�(�)k+(�))
�1(iI1 � k�(�); I1 � ik�(�));
(4.37)
and inequalities (4.12) in Th. 4.2 are equivalent to
Z
J�
�
I1 � ik�(�)
�iI1 + k�(�)
��
X�
�
(t)w�(t)X�(t)
�
I1 � ik�(�)
�iI + k�(�)
�
dt
�
1
=�
(I1 � k��(�)k�(�)): (4.38)
If one passes in (4.37), (4.38) from strict contractions k�(�) to the Nevanlinna
operator functions
�m�(�) = �(�iI1 + k�(�))(I1 � ik�(�))
�1; �=m�(�) >> 0;
and takes into account that
m�(�)�m+(�) = 2i(�I1 + ik�(�))
�1(I1 � k�(�)k+(�))(I1 + ik+(�))
�1;
then projection (4.37) acquires the following form known from [1]:
P(�) =
�
I1
m+(�)
�
(m�(�)�m+(�))
�1(m�(�);�I1); (4.39)
468 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Characteristic Operators and Projections and on the Solutions of Weyl...
and inequalities (4.38) becomeZ
J�
��(t; �)w�(t) �(t; �)dt � �
=m�(�)
=�
; (4.40)
with
�(t; �) =
�
x1(t; �) + x2(t; �)m�(�)
x3(t; �) + x4(t; �)m�(�)
�
: (4.41)
In a similar way, the projection P (4.25) in Th. 4.4 is given by
P(�) =
1
2i
�
I1
m+(�)
�
�(I1 + ik+(�))(I1 � u�(�)k+(�))
�1(iI1 � u�(�); I1 � iu�(�)); (4.42)
with u�(�) being the contraction in H1 that depends analytically on �.
Take into account that
2i(I1 � u�(�)k+(�))(I1 + ik+(�))
�1 = iI1 � u�(�) + (I1 � iu�(�))m+(�)
and denote u(�) = �iu�(�)
a1(�) = �i(u(�) + I1); a2 = u(�)� I1; (4.43)
to observe that the projection P(�) (4.25), (4.42) acquires the form
P(�) =
�
I1
m+(�)
�
(a2(�)� a1(�)m+(�))
�1(a2(�);�a1(�)): (4.44)
Thus in boundary condition (4.26) we can set
M� =
�
a1(�) 0
a2(�) 0
�
:
We follow terminology of [13]�[17] by calling the operators m�(�) 2 B(H1)
and m+(�) 2 B(H1) that depend analytically on � and satisfy (4.40), (4.41),
the Weyl functions of (0.1), (4.36) on (a; c) and (c; b), respectively.
Formula (4.39) indicates that de�nition of the Weyl functions in this work is
equivalent to that of [1]?.
?In [1] m�(�) are denoted by n�(�).
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 469
V.I. Khrabustovsky
Remark 4.5. (In the case dimH < 1 cf. [8]). Suppose the interval (a; b)
is �nite. Then the operators m�(�) 2 B(H1) and m+(�) 2 B(H1) are the Weyl
functions of (0.1), (4.36) on (a; c) and on (c; b), respectively if and only if they
admit the representation
m�(�) = (x�1(a;
��)a2(�)� x�3(a;
��)a1(�))(x
�
4(a;
��)a1(�)� x�2(a;
��)a2(�))
�1;
(4.45)
m+(�) = (x�1(b;
��)b2(�)� x�3(b;
��)b1(�))(x
�
4(b;
��)b1(�)� x�2(b;
��)b2(�))
�1; (4.46)
with xj(t; �) 2 B(H1) see (4.33),
a1(�) = �i(U(�) + I1); a2 = U(�)� I1; (4.47)
b1(�) = V (�)� I1; b2 = �i(V (�) + I1); (4.48)
U(�), V (�) being arbitrary contractions in H1 (the latter assumption guaranties
that in (4.45), (4.46) one has (:::)�1 2 B(H1)) that depend analytically on �.
In this setting solution of (0.1), (4.36) x�(t) (1.9), (1.20), (4.39), (4.45),
(4.46) is a solution of the boundary-value problem (0.1), (1.14) with
M� =
�
a1(�) 0
a2(�) 0
�
; N� =
�
0 b1(�)
0 b2(�)
�
;
aj(�), bj(�) 2 B(H1), see (4.47), (4.48).
Remark 4.6. 10 (cf. Remark 4.1). One can choose the contraction u(�) in
(4.43) so that in (4.43) one has a�11 2 B(H1), and a2(�)a
�1
1 (�) = m�(�) is an
arbitrary Weyl function of (0.1), (4.36) on (a; c). Under this choice of u(�) the
projections (4.44), (4.39) coincide, hence M(�) (1.20), (4.44) is also a charac-
teristic operator of (0.1), (4.36) on (a; b).
20. If in (4.43) u(�) is unitary (hence by [35, p. 210] u(�) = u does not
depend on �), then the operators a1, a2 admit the representation (cf. [33])
a1 = cos� �K; a2 = sin� �K;
with � = �� 2 B(H1);K = �2ie�i�, and the projection (4.44) can be written in
the form
P(�) =
�
I1
m+(�)
�
(sin�� cos�m+(�))
�1(sin�;� cos�)
with (:::) 2 B(H1).
In de�nition (1.9), (1.10) of c.o. (0.1), (4.36) the operator solution X�(t) is
often replaced by
X�(t; �) = X�(t)
�
sin� cos �
� cos � sin�
�
;
with � = �� 2 B(H1).
470 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Characteristic Operators and Projections and on the Solutions of Weyl...
Remark 4.7. (In the case dimH = 2 cf. [17]). Suppose that the operators
�; � commute. Then, if one replaces in the de�nition of c.o. of (0.1), (4.36)
the operator solution X�(t) by X�(t; �), the characteristic projection P(�) from
n020 of Remark 4.6 turns into the characteristic projection
P�(�) =
�
I1
m+(�; �)
�
�(sin(�� �) + cos(�� �)m+(�; �))
�1(cos(�� �); sin(�� �)); (4.49)
with a new Weyl function m+(�; �) being related to the Weyl function m+(�)
as follows:
m+(�; �) = (cos � + sin�m+(�))(sin� � cos�m+(�))
�1 (4.50)
(in (4.49), (4.50) one has (:::)�1 2 B(H)).
We are about to demonstrate that the Weyl functions m�(�) are analogues of
the Weyl functions of Dirac and Sturm�Liouville equations, as well as analogues
for characteristic matrix of a scalar symmetric di�erential operator of even order
on the semiaxis.
In the case of the Dirac type homogeneous equation (0.1), (4.36) when the
weight =w�(t) = I, i.e., for the equation�
0 �I1
I1 0
�
x0(t) +H0(t)x(t) = �x(t); (4.51)
with H0(t) = H�
0 (t) 2 B(H), the inequalities (4.40) are equivalent to the inequa-
lities as follows:Z
J�
�
(x1(t; �) + x2(t; �)m�(�))
�(x1(t; �) + x2(t; �)m�(�))
+(x3(t; �) + x4(t; �)m�(�))
�(x3(t; �) + x4(t; �)m�(�))
�
dt � �
=m�(�)
=�
; (4.52)
i.e., m�(�) (or m
�1
� (�)) are operator analogues of the scalar Weyl functions for
the Dirac equation (see, e.g., [13], [17]).
One has a di�erent but equivalent to (4.51) form of the Dirac equation, which
is commonly used:
i
�
I1 0
0 �I1
�
y0(t) + ~H0(t)y(t) = �y(t); (4.53)
with ~H0(t) = 2S��1H0(t)S
�1, S =
�
I1 iI1
iI1 I1
�
.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 471
V.I. Khrabustovsky
In this case inequalities (4.52) are equivalent to the claim that for the operator
solutions
�(t; �) = F (t) +G(t)m�(�) 2 B(H1;H
2
1)
of (4.53), where F (t); G(t) 2 B(H1;H
2
1) are such operator solutions of (4.53) that
fF (0); G(0)g = S, i.e., F (0) =
�
I1
iI1
�
; G(0) =
�
iI1
I1
�
2 B(H1;H
2
1), the following
inequalities hold Z
J�
��(t; �) �(t; �)dt � �
2=m�(�)
=�
:
Consequently, m�(�) are also operator analogues of the Weyl functions
for the Dirac equation of the form (4.53), considered for the case dimH = 2 in
the work by V.A. Marchenko [15].
In the case when H1 = Hn
2 and the weight given by w�(t) = diag(I2; O2; ::; O2)
(in particular, a symmetric equation of an arbitrary even order 2n in H2 (e.g.,
the Sturm�Liouville equation) reduces to equation (0.1), (4.36) with such weight),
inequalities (4.40) are equivalent toZ
J�
[(x1(t; �); : : : ; xn(t; �)) + (xn+1(t; �); : : : ; x2n(t; �))m�(�)]
�
�[(x1(t; �); : : : ; xn(t; �)) + (xn+1(t; �); : : : ; x2n(t; �))m�(�)]dt � �
=m�(�)
=�
;
with xj(t; �) = x1j(t; �) being the �rst line operator elements of the operator
matrix X�(t) = (xij(t; �))
2n
i;j=1; xij(t; �) 2 B(H2). That is, the operators m�(�)
(or m�1
� (�)) are analogues of the Weyl functions of the Sturm�Liouville equation
[13]�[15], [17], as well as an operator analogue of characteristic matrix [40] for the
scalar symmetric equation of even order 2n on the semiaxis.
III. Let
H = H1 �H1; Q(t) =
�
0 I1
I1 0
�
: (4.54)
This case reduces to the case II since�
I1 0
0 iI1
��
0 iI1
�iI1 0
��
I1 0
0 �iI1
�
=
�
0 I1
I1 0
�
:
In particular, (4.40), (4.46) in the case (4.54) turns to the inequalities obtained
for the case dimH < 1 in [16, p. 337] for the Weyl type solutions of equation
x0(t) = i�
�
0 I1
I1 0
�
H(t)x(t) with t 2 I+, 0 � H(t) 2 B(H),
R
l
c
H(t)dt >> 0 for
some l 2 �I+:
472 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4
On the Characteristic Operators and Projections and on the Solutions of Weyl...
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Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 473
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