Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions
UDC 517.9We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function $Q$ represented by a self-adjoint linear relation $A$ in a Pontryagin space is decomposed by means of the reducing subspaces of $A.$...
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Institute of Mathematics, NAS of Ukraine
2022
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512265941286912 |
|---|---|
| author | Borogovac, M. Borogovac, M. |
| author_facet | Borogovac, M. Borogovac, M. |
| author_sort | Borogovac, M. |
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| datestamp_date | 2022-10-24T09:23:11Z |
| description | UDC 517.9We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function $Q$ represented by a self-adjoint linear relation $A$ in a Pontryagin space is decomposed by means of the reducing subspaces of $A.$ The sum of two functions $Q_{i}{\in N}_{\kappa_{i}}(\mathcal{H}),$ $i=1, 2,$ minimally represented by the triplets $(\mathcal{K}_{i},A_{i},\Gamma_{i})$ is also studied. For this purpose, we create a model $(\tilde{\mathcal{K}},\tilde{A},\tilde{\Gamma })$ to represent $Q:=Q_{1}+Q_{2}$ in terms of $(\mathcal{K}_{i},A_{i},\Gamma_{i})$. By using this model, necessary and sufficient conditions for $\kappa =\kappa_{1}+\kappa_{2}$ are proved in the analytic form. Finally, we explain how degenerate Jordan chains of the representing relation $A$ affect the reducing subspaces of $A$ and the decomposition of the corresponding function $Q.$ |
| doi_str_mv | 10.37863/umzh.v74i7.6084 |
| first_indexed | 2026-03-24T03:26:03Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i7.6084
UDC 517.9
M. Borogovac (Boston Mutual Life Insurance Company, Canton, MA, USA)
REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS
AND APPLICATION TO GENERALIZED NEVANLINNA FUNCTIONS
ЗВIДНIСТЬ САМОСПРЯЖЕНИХ ЛIНIЙНИХ СПIВВIДНОШЕНЬ
I ЗАСТОСУВАННЯ ДО УЗАГАЛЬНЕНИХ ФУНКЦIЙ НЕВАНЛIННИ
We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a
generalized Nevanlinna function Q represented by a self-adjoint linear relation A in a Pontryagin space is decomposed by
means of the reducing subspaces of A. The sum of two functions Qi\in N\kappa i(\scrH ), i = 1, 2, minimally represented by the
triplets (\scrK i, Ai,\Gamma i) is also studied. For this purpose, we create a model ( \~\scrK , \~A, \~\Gamma ) to represent Q := Q1 + Q2 in terms
of (\scrK i, Ai,\Gamma i). By using this model, necessary and sufficient conditions for \kappa = \kappa 1 + \kappa 2 are proved in the analytic form.
Finally, we explain how degenerate Jordan chains of the representing relation A affect the reducing subspaces of A and
the decomposition of the corresponding function Q.
Наведено необхiднi та достатнi умови звiдностi самоспряженого лiнiйного спiввiдношення у просторi Крейна. Далi
узагальнена функцiя Неванлiнни Q, що представлена самоспряженим лiнiйним спiввiдношенням A у просторi Пон-
трягiна, розкладається за допомогою звiдних пiдпросторiв A. Також вивчається сума двох функцiй Qi\in N\kappa i(\scrH ),
i = 1, 2, мiнiмально представлена трiйками (\scrK i, Ai,\Gamma i). З цiєю метою створено модель ( \~\scrK , \~A, \~\Gamma ), що представляє
Q := Q1+Q2 в термiнах (\scrK i, Ai,\Gamma i). За допомогою цiєї моделi необхiднi та достатнi умови для \kappa = \kappa 1+\kappa 2 дове-
дено в аналiтичнiй формi. Насамкiнець ми пояснюємо, яким чином виродженi жордановi ланцюги представницьких
спiввiдношень A впливають на звiднi пiдпростори A та на розклад вiдповiдної функцiї Q.
1. Preliminaries and introduction 1.1. Preliminaries. Let N, R, and C denote sets of positive
integers, real numbers, and complex numbers, respectively. Let (., .) denote the (definite) scalar
product in the Hilbert space \scrH , and let \scrL (\scrH ) denote the space of bounded linear operators in \scrH .
Definition 1.1. An operator-valued complex function Q : \scrD (Q) \rightarrow \scrL (\scrH ) belongs to the class of
generalized Nevanlinna functions N\kappa (\scrH ) if it satisfies the following requirements:
Q is meromorphic in C\setminus R,
Q(\=z)\ast = Q(z), z \in \scrD (Q),
and
the Nevanlinna kernel NQ(z, w) :=
Q(z) - Q(w)\ast
z - \=w
, z, w \in \scrD (Q) \cap C+,
has \kappa negative squares. In other words, for arbitrary n \in N, z1, . . . , zn \in \scrD (Q) \cap C+ and
h1, . . . , hn \in \scrH , the Hermitian matrix
\bigl(
NQ(zi, zj)hi, hj
\bigr) n
i,j=1
has \kappa negative eigenvalues at most,
and, for at least one choice of n; z1, . . . , zn, and h1, . . . , hn, it has exactly \kappa negative eigenvalues.
It is easy to verify that Nevanlinna kernel is a Hermitian kernel, i.e., NQ(z, w)
\ast = NQ(w, z),
z, w \in \scrD (Q) \cap C+.
The following definitions of a linear relation and basic concepts related to it can be found, for
example, in [1, 4, 18]. In the sequel, \scrH , \scrK , and \scrM are inner product spaces. Recall, a set M is
called linear manifold (or linear space) if, for any two vectors x, y \in M and for any two scalars
\alpha , \beta \in C, it holds \alpha x + \beta y \in M. A linear relation from \scrH into \scrK is a linear manifold T of the
product space \scrH \times \scrK . If \scrH = \scrK , T is said to be a linear relation in \scrK . We will use the following
concepts and notations for linear relations, T and S from \scrH into \scrK and a linear relation R from \scrK
c\bigcirc M. BOROGOVAC, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7 893
894 M. BOROGOVAC
into \scrM :
D(T ) :=
\bigl\{
f \in \scrH | \{ f, g\} \in T for some g \in \scrK
\bigr\}
,
R(T ) :=
\bigl\{
g \in \scrK | \{ f, g\} \in T for some f \in \scrH
\bigr\}
,
\mathrm{k}\mathrm{e}\mathrm{r}T :=
\bigl\{
f \in \scrH | \{ f, 0\} \in T
\bigr\}
,
T (0) :=
\bigl\{
g \in \scrK | \{ 0, g\} \in T
\bigr\}
,
T (f) :=
\bigl\{
g \in \scrK | \{ f, g\} \in T\} , f \in D(T ),
T - 1 :=
\bigl\{
\{ g, f\} \in \scrK \times \scrH | \{ f, g\} \in T
\bigr\}
,
zT :=
\bigl\{
\{ f, zg\} \in \scrH \times \scrK | \{ f, g\} \in T
\bigr\}
, z \in C,
S + T :=
\bigl\{
\{ f, g + k\} | \{ f, g\} \in S, \{ f, k\} \in T
\bigr\}
,
RT :=
\bigl\{
\{ f, k\} \in \scrH \times \scrM | \{ f, g\} \in T, \{ g, k\} \in R for some g \in \scrK
\bigr\}
,
T+ :=
\bigl\{
\{ k, h\} \in \scrK \times \scrH | [k, g] = (h, f) for all \{ f, g\} \in T
\bigr\}
,
T\infty :=
\bigl\{
\{ 0, g\} \in T
\bigr\}
.
Note that in definition of the adjoint linear relation T+, we use the following notation for inner
product spaces: (\scrH , (., .)) and (\scrK , [., .]).
If \mathrm{m}\mathrm{u}\mathrm{l}T := T (0) = \{ 0\} , we say that T is an operator, or single-valued linear relation. A linear
relation is closed if it is a closed subset in the product space \scrH \times \scrK .
Let A be a linear relation in \scrK . We say that A is symmetric (self-adjoint) if it holds A \subseteq A+
(A = A+). Every point \alpha \in C for which \{ f, \alpha f\} \in A, with some f \not = 0, is called a finite
eigenvalue. The corresponding vectors are eigenvectors belonging to the eigenvalue \alpha . The set that
consists of all points z \in C for which the relation (A - zI) - 1 is an operator defined on the entire
\scrK , is called the resolvent set \rho (A).
Let \kappa \in N \cup \{ 0\} and (\scrK , [., .]) denote a Krein space. That is, a complex vector space on which
a scalar product, i.e., a Hermitian sesquilinear form [., .], is defined such that the decomposition
\scrK = \scrK + \.+\scrK -
of \scrK exists, where
\bigl(
\scrK +, [., .]
\bigr)
and
\bigl(
\scrK - , - [., .]
\bigr)
are Hilbert spaces which are mutually orthogo-
nal with respect to the form [., .]. Every Krein space
\bigl(
\scrK , [., .]
\bigr)
is associated with a Hilbert space\bigl(
\scrK , (., .)
\bigr)
, which is defined as a direct and orthogonal sum of the Hilbert spaces
\bigl(
\scrK +, [., .]
\bigr)
and\bigl(
\scrK - , - [., .]
\bigr)
. The topology in a Krein space \scrK is the topology of the associated Hilbert space\bigl(
\scrK , (., .)
\bigr)
. For properties of Krein spaces see, e.g., [5] (Chapt. V).
If the scalar product [., .] has \kappa (< \infty ) negative squares, then we call it a Pontryagin space of
the index \kappa . The definition of a Pontryagin space and other concepts related to it can be found, e.g.,
in [11].
The following construction of a Pontryagin space can be found in [9, 10, 12] and a similar
construction of a Hilbert space can be found in [14]:
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REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 895
For any generalized Nevanlinna function Q, a linear space L(Q) with a (possibly degenerate)
indefinite inner product [., .] can be introduced as follows:
Consider the set of all finite formal sums\sum
\varepsilon zhz, z \in \scrD (Q),
where hz \in \scrH , and \varepsilon z is a symbol associated with each z \in \scrD (Q). Then an inner product is defined
by
[\varepsilon zhz, \varepsilon whw] :=
\biggl(
Q(z) - Q ( \=w)
z - \=w
hz, hw
\biggr)
, z, w \in \scrD (Q), z \not = \=w, hz, hw \in \scrH ,
[\varepsilon zhz, \varepsilon \=zh\=z] :=
\bigl(
Q\prime (z)hz, h\=z
\bigr)
, z \in \scrD (Q).
To ease communication, let us call L(Q) the state manifold of Q. The linear relation defined by
A0 := \mathrm{l}.\mathrm{s}.
\Biggl\{ \Biggl\{ \sum
s
\varepsilon zshs,
\sum
s
zs\varepsilon zshs
\Biggr\}
:
\sum
s
hs = 0, zs \in \scrD (Q)
\Biggr\}
is symmetric. For z0 \in \scrD (Q), the operator \Gamma z0 : \scrH \rightarrow L(Q) is defined by \Gamma z0h = \varepsilon z0h. The
Pontryagin space \scrK is obtained by factorization of L(Q) with respect to its isotropic part L0
0 :=
:= L(Q) \cap L(Q) [\bot ] and by completion of the factor space. It is called the state space of Q. In the
process, A0 and \Gamma z0 give rise to the self-adjoint relation A in \scrK and bounded linear operator \Gamma :
\scrH \rightarrow \scrK , with z0 \in \rho (A). Then the following theorem holds.
Theorem 1.1. A function Q : \scrD (Q) \rightarrow L(\scrH ) is a generalized Nevanlinna function of the index
\kappa , denoted by Q \in N\kappa (\scrH ), if and only if it has a representation of the form
Q(z) = Q(z0)
\ast + (z - \=z0)\Gamma
+
\bigl(
I + (z - z0) (A - z) - 1
\bigr)
\Gamma , z \in \scrD (Q), (1.1)
where A is a self-adjoint linear relation in some Pontryagin space (\scrK , [., .]) of the index \~\kappa \geq \kappa ; \Gamma :
\scrH \rightarrow \scrK is a bounded operator. (Obviously \rho (A) \subseteq \scrD (Q)). This representation can be chosen to be
minimal, that is,
\scrK = \mathrm{c}.\mathrm{l}.\mathrm{s}.
\bigl\{
\Gamma zh : z \in \rho (A), h \in \scrH
\bigr\}
,
where
\Gamma z :=
\bigl(
I + (z - z0)(A - z) - 1
\bigr)
\Gamma .
If realization (1.1) is minimal, then Q \in N\kappa (\scrH ) if and only if \~\kappa equals \kappa . In the case of minimal
representation \rho (A) = D(Q) and the triple (\scrK , A,\Gamma ) is uniquely determined (up to isomorphism).
Such operator representations were developed by M. G. Krein and H. Langer [12, 13] and later
converted to representations in terms of linear relations (see, e.g., [9, 10]).
In this paper, a point \alpha \in C is called a generalized pole of Q if it is an eigenvalue of the
representing relation A. It may be an isolated singularity, i.e., an ordinary pole, as well as an
embedded singularity of Q. The latter may be the case only if \alpha \in R.
1.2. Introduction. We start Section 2 with extending the definition of reducibility of operators in
Hilbert spaces to reducibility of linear relations in Krein spaces. Then in Lemma 2.2 we prove several
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
896 M. BOROGOVAC
statements about decompositions, i.e., about relation matrix, of a linear relation in a Krein space \scrK
that we need in the proof of the main result, Theorem 2.1. In that theorem we give necessary and
sufficient conditions for a self-adjoint linear relation A in \scrK to be reduced to the sum A = A1 [+]A2,
where [+] is direct and orthogonal sum of linear relations, Ai are self-adjoint linear relations in the
reducing subspaces \scrK i, and \scrK = \scrK 1 [\dotplus ]\scrK 2. Then, by means of reducing subspaces and reducing
linear relations, we study decompositions of a generalized Nevanlinna function Q.
The number of negative squares \kappa \in N \cup \{ 0\} is an important feature of the generalized Nevan-
linna function Q. Recall that, if functions Qi, i = 1, 2, satisfy
(i) Qi\in N\kappa i(\scrH ), 0 \leq \kappa i, i = 1, 2,
(ii) Q(z) = Q1(z) +Q2(z),
then Q belongs to some generalized Nevanlinna class N\kappa (\scrH ) and \kappa \leq \kappa 1 + \kappa 2 holds.
There are two basic questions:
(a) Given function Q\in N\kappa (\scrH ), under what conditions does there exist a decomposition Q(z) =
= Q1(z) +Q2(z), Qi\in N\kappa i(\scrH ), i = 1, 2, that satisfies \kappa = \kappa 1 + \kappa 2?
(b) Given two functions Qi\in N\kappa i(\scrH ), i = 1, 2, is the number of negative squares preserved in
the sum Q = Q1 +Q2 or not?
In other words, we investigate the circumstances under which functions Q, Q1 and Q2 that
satisfy (i) and (ii) also satisfy
(iii) \kappa 1 + \kappa 2 = \kappa .
The question of preservation of the number of negative squares of the sum of Hermitian kernels
K(z, w) = K1(z, w) + K2(z, w) was studied in [3]. The authors give necessary and sufficient
conditions for \kappa 1 + \kappa 2 = \kappa in terms of complementary reproducing kernel Pontryagin spaces \scrK 1,
\scrK 2, c.f. [3] (Theorem 1.5.5). We alternatively give necessary and sufficient conditions for \kappa 1+\kappa 2 = \kappa
in terms of triplets (\scrK i, Ai,\Gamma i), i = 1, 2, associated with minimal representations of the form (1.1),
c.f. Theorem 3.2.
The question of preservation of the number of negative squares in products, sums, and in some
transformations of generalized Nevanlinna functions has been, among other topics, summarised in
the survey [15]. In the present paper, we prove analytic criteria that establish whether the sum of the
indexes of the functions that comprise the sum is equal or it is greater than the negative index of the
sum.
It is very difficult to determine the negative index \kappa of a given generalized Nevanlinna function.
The established relation between negative indexes of the above sum (ii) gives us information that
might help in determining the numbers of negative indexes of the functions in the sum.
There are interesting results about decompositions of generalized Nevanlinna functions in [8, 13],
for matrix and scalar functions represented by unitary and self-adjont operators. In those papers, the
decompositions of Q \in Nn\times n
\kappa were obtained by means of spectral families of the representing
operators and their appropriate invariant spectral subspaces. The decomposing functions Qi obtained
by that method must have disjoint sets of generalized poles (see [8], Proposition 3.1). In the present
article, we do not use spectral families and spectral subspaces; we use instead a concept of the
reducing subspaces of the representing self-adjoint relation in the Pontryagin state space. That way
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 897
we obtain decomposition where decomposing functions Qi, i = 1, 2, may have common generalized
poles.
In Theorem 3.1, we give a general answer on the question (a); we decompose function Q by
means of reducing subspaces \scrK i and reducing relations Ai of the representing relation A.
Regarding sums of generalized Nevanlinna functions, in [8] (Proposition 3.2) it has been proven
that the sum of two generalized Nevanlinna matrix functions preserves the number of negative squares
under the condition that functions in the sum have disjoint sets of generalized poles. In our study we
do not use that condition.
We start the study of the sum Q := Q1 + Q2 with two functions Qi \in N\kappa i(\scrH ), i = 1, 2,
represented minimally in Pontryagin spaces by triplets (\scrK i, Ai,\Gamma i), i = 1, 2. Then we create Pon-
tryagin space \~\scrK := \scrK 1 [\dotplus ]\scrK 2, and representation of the function Q := Q1 + Q2 in terms of the
triplets (\scrK i, Ai,\Gamma i). That representation, denoted by (3.3) in the text, we call orthogonal sum repre-
sentation. Then, in Theorem 3.2, we describe the structure of the possibly nonminimal state space
\~\scrK := \scrK 1 [\dotplus ]\scrK 2 representing the sum Q = Q1 + Q2. In Corollary 3.1, we give necessary and
sufficient conditions for \kappa = \kappa 1 + \kappa 2 in terms of the inner structure of the state space \~\scrK .
In Theorems 4.1 and 4.2, we prove some analytic criteria for \kappa = \kappa 1 + \kappa 2 or \kappa < \kappa 1 + \kappa 2.
These criteria are easy to use; we do not need to know operator representations of the functions
comprising the sum. Given how Definition 1.1 is impractical for use and how difficult it is to find
operator representations, our criteria are useful tool for research of both, the underlying state space,
and features of the sum Q := Q1 +Q2.
In Proposition 5.1, we decompose a function Q by means of Theorem 3.1 using linear spans
of nondegenerate Jordan chains as reducing subspaces. Proposition 5.1 is a straightforward result
that we needed to approach the more complicated case of degenerate chains which we study in
Proposition 5.2. In Proposition 5.2, we consider the model where the self-adjoint operator A in a
Pontryagin space \scrK has two simple, independent, and degenerate chains (neutral eigenvectors) at
\alpha \in R. We prove that, unlike nondegenerate chains, studied in Proposition 5.1, the two degenerate
chains at \alpha \in R cannot reduce the representing operator and cannot induce two different functions
Qi in any decomposition of Q. The conclusion of Section 5 is in Corollary 5.1.
2. Reducing subspaces of the self-adjoint linear relation in the Krein space. In the sequel
[+], rather than [\dotplus ], denotes direct and orthogonal sum of both, relations and vectors. From the
context it is usually clear when we deal with “operator-like” addition of linear relations, as well as
when we deal with addition of relations as subspaces, and addition of vectors. If necessary, we will
specify.
Lemma 2.1. Assume that \scrK 1 and \scrK 2 are Krein spaces and Al \subseteq \scrK 2
l , l = 1, 2, are linear
relations. We can define direct orthogonal sum
\scrK := \scrK 1 [+]\scrK 2
and
A = A1 [+]A2 :=
\Biggl\{ \Biggl(
hi [+]hj
h\prime i [+]h\prime j
\Biggr)
:
\Biggl(
hl
h\prime l
\Biggr)
\in Al, l = 1, 2
\Biggr\}
\subseteq \scrK 2.
The linear relation A := A1 [+]A2 is symmetric (self-adjoint) in \scrK if and only if linear relations
Al \subseteq \scrK 2
l , l = 1, 2, are symmetric (self-adjoint).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
898 M. BOROGOVAC
Proof. This lemma is a straightforward verification and left to the reader.
Let A be a linear relation in the Krein space \scrK ,
\scrK := \scrK 1 [+]\scrK 2,
where nontrivial subspaces \scrK l are also Krein spaces and El : \scrK \rightarrow \scrK l, l = 1, 2, are the corresponding
orthogonal projections. The following four linear relations can be introduced:
Aj
i :=
\Biggl\{ \Biggl(
hi
hji
\Biggr)
: hi \in D(A) \cap \scrK i, h
j
i \in EjA(hi)
\Biggr\}
\subseteq \scrK i \times \scrK j , i, j = 1, 2.
In this notation the subscript i is associated with the domain subspace \scrK i, the superscript j is
associated with the range subspace \scrK j . For example
\biggl(
h1
h21
\biggr)
\in A2
1.
Let us now extend the definition of the reducing subspaces of the unbounded operator in the
Hilbert space, (see, e.g., [2], Section 40), to the reducing subspaces of the (multivalued) linear
relation in Krein space.
Definition 2.1. Let (\scrK , [., .]) be a Krein space, \scrK 1 \subset \scrK be a nontrivial Krein subspace of \scrK ,
and \scrK 2 = \scrK [ - ]\scrK 1. We will say that the subspaces \scrK 1 and \scrK 2 reduce relation A if there exist
linear relations Ai \subseteq \scrK i \times \scrK i, i = 1, 2, such that it holds
A = A1 [+]A2,
where [+] stands for direct orthogonal addition of relations, as defined in Lemma 2.1. The relations
Ai are called reducing relations of A.
Recall, if \scrK is a Pontryagin space and \scrK 1 is a nondegenerate closed subspace, then \scrK =
= \scrK 1 [+]\scrK 2 and both \scrK i, i = 1, 2, are also Pontryagin spaces, see [11] (Theorem 3.2 and Corol-
lary 2).
Lemma 2.2. Let A, Aj
i , \scrK i, Ei; i, j = 1, 2, be introduced as above. If for either of orthogonal
projections Ei : \scrK \rightarrow \scrK i, i = 1, 2, it holds Ei(D(A)) \subseteq D(A), then:
(i) E1
\bigl(
A(0)
\bigr)
= A1
1(0) = A1
2(0), E2
\bigl(
A(0)
\bigr)
= A2
1(0) = A2
2(0);
(ii) A =
\bigl(
A1
1 +A2
1
\bigr)
\^+
\bigl(
A1
2 +A2
2
\bigr)
, where + stands for operator-like addition, and \^+ stands for
addition of the subspaces, not necessarily direct;
(iii) if B : \scrK i \rightarrow \scrK j , i, j = 1, 2, is a closed relation, then
B(0) = D(B[\ast ])[\bot ](\subseteq \scrK j); (2.1)
(iv) If A is symmetric, then it holds A1
2 \subseteq A2
1
[\ast ]
, A2
1 \subseteq A1
2
[\ast ]
, A1
1 \subseteq A1
1
[\ast ]
, A2
2 \subseteq A2
2
[\ast ]
;
(v) if A is symmetric and D(A) \cap \scrK i is dense in \scrK i, then Ai
i is single-valued relation and
A(0) \subseteq \scrK j , i.e., A(0) = Aj
i (0) = Aj
j(0), j \not = i, i, j = 1, 2.
Proof. Note that from Ei(D(A)) \subseteq D(A) it follows Ej
\bigl(
D(A)
\bigr)
\subseteq D(A), i \not = j, and
Ei(D(A)) = \scrK i \cap D(A), i = 1, 2.
Then the first two statements of the lemma follow directly from the definition of the relations Aj
i .
(iii) If B is a linear relation in a Krein space, not necessarily closed, then it holds
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REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 899
B(0) \subseteq D
\Bigl(
B[\ast ]
\Bigr) [\bot ]
.
Indeed, y \in B(0) \Rightarrow
\biggl(
0
y
\biggr)
\in B \Rightarrow [y, k] = 0 \forall
\biggl(
k
k\prime
\biggr)
\in B[\ast ] \Rightarrow B(0) \subseteq D
\Bigl(
B[\ast ]
\Bigr) [\bot ]
.
To prove the converse inclusion (\supseteq ) we need assumption that B is closed. Then we have
y \in D
\Bigl(
B[\ast ]
\Bigr) [\bot ]
\Rightarrow [y, k] = 0 \forall
\Biggl(
k
k\prime
\Biggr)
\in B[\ast ] \Rightarrow
\Biggl(
0
y
\Biggr)
\in B[\ast ][\ast ] = \=B = B \Rightarrow y \in B(0).
Hence, the converse inclusion holds too, which completes the proof of (2.1).
(iv) Let us here clarify notation that we will frequently use in this lemma and the next theorem.
For hi \in D(A) \cap \scrK i it holds\Biggl(
hi
h\prime i
\Biggr)
\in A \leftrightarrow
\Biggl(
hi
h\prime i
\Biggr)
=
\biggl(
hi
h1i [+]h2i
\biggr)
\in A1
i +A2
i , i = 1, 2,
where h\prime i = h1i [+]h2i \in \scrK 1 [+]\scrK 2 and A1
i +A2
i is operator-like sum. For h = h1 [+]h2 it holds\Biggl(
h
h\prime
\Biggr)
\in A \leftrightarrow
\Biggl(
h
h\prime
\Biggr)
=
\Biggl(
h1[+]h2
h11 [+]h21 + h12 [+]h22
\Biggr)
,
\Biggl(
hi
hji
\Biggr)
\in Aj
i , i, j = 1, 2.
In the sequel we will for addition of vectors frequently use simply + rather than [+] because the
notation of the vectors in the particular sums indicate when the direct orthogonal sum applies.
Let us now assume that A is a symmetric relation and let us, for example, show that it holds
A1
2 \subseteq A2
1
[\ast ]
.
Let us select arbitrary
\biggl(
h2
h12
\biggr)
\in A1
2. Then, for every
\biggl(
h1
h21
\biggr)
\in A2
1, there exist
\biggl(
h2
h12 + h22
\biggr)
\in A and\biggl(
h1
h11 + h21
\biggr)
\in A. Because A is symmetric, it holds
[h1, h
1
2 + h22] = [h11 + h21, h2].
Hence,
[h1, h
1
2] = [h21, h2].
This proves
\biggl(
h2
h12
\biggr)
\in A2
1
[\ast ]
, i.e., A1
2 \subseteq A2
1
[\ast ]
.
By the same token it holds:
Aj
i \subseteq Ai
j
[\ast ]
, i, j = 1, 2.
(v) We will prove this statement for i = 1, j = 2. Hence, we assume that D(A) \cap \scrK 1 is dense
in \scrK 1. Let us apply formula (2.1) on the (closed) relation B = A1
1
[\ast ]
. We get B(0) = A1
1
[\ast ]
(0) =
= D( \=A1
1)
[\bot ]
= D(A1
1)
[\bot ]
= \{ 0\} . Then it follows
A1
1(0) \subseteq A1
1
[\ast ]
(0) = D(A1
1)
[\bot ]
= \{ 0\} \Rightarrow
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900 M. BOROGOVAC
\Rightarrow A(0) = E2A(0) = A2
1(0) = A2
2(0) \subseteq \scrK 2.
In the following theorem, we give necessary and sufficient conditions for a self-adjoint linear
relation in a Krein space to be reduced in the sense of Definition 2.1. The important statement
is (vii). Some of the other listed statements are merely the important steps in the proof of the
statement (vii).
Theorem 2.1. Assume that A is a self-adjoint linear relation in a Krein space
\bigl(
\scrK , [., .]
\bigr)
,
\scrK 1 \subset \scrK is a nontrivial nondegenerate subspace, and \scrK 2 is the orthogonal complement of \scrK 1
in \scrK , i.e.,
\scrK = \scrK 1 [+]\scrK 2.
If it holds E1(D(A)) \subseteq D(A) and A(\scrK 1 \cap D(A)) \subseteq \scrK 1, then:
(i) A = A1
1
\^+
\bigl(
A1
2 +A2
2
\bigr)
,
(ii) A2
2 is single-valued self-adjoint relation in \scrK 2,
(iii) A2
2 and A1
2 are densely defined operators in \scrK 2,
(vi) A(0) = A1
1(0) = D
\Bigl(
A1
1
[\ast ]
\Bigr) [\bot ]
= D(A)[\bot ],
(v) A1
2
[\ast ] is single valued,
(vi) A1
1 = A1
1
[\ast ] \leftrightarrow R(A1
2) \subseteq A1
1(0),
(vii) A = A1
1 [+]A2
2 if and only if A1
1 is self-adjoint,
(viii) if A(D(A) \cap \scrK 1) \subseteq \scrK 1 is dense in \scrK 1, then A1
1
[\ast ]
= A1
1 is operator as well.
Proof. By assumption h21 \equiv 0 \forall h1 \in \scrK 1 \cap D(A). Then the statement (i) follows from
Lemma 2.2 (ii).
(ii) Because A(\scrK 1 \cap D(A)) \subseteq \scrK 1 it holds A(0) \subseteq \scrK 1. Hence, E2A(0) = A2
2(0) = \{ 0\} , i.e., A2
2
is single-valued. Let us now prove that A2
2 is a self-adjoint operator. Assume that
\biggl(
k2
k22
\biggr)
\in A2
2
[\ast ]
.
We will first verify that for every
\biggl(
h1 + h2
h11 + h12 + h22
\biggr)
\in A it holds
\bigl[
k2, h
1
1 + h12 + h22
\bigr]
=
\bigl[
k22, h1 + h2
\bigr]
.
This equation is obviously equivalent to\bigl[
k2, h
2
2
\bigr]
=
\bigl[
k22, h2
\bigr]
,
which holds according to assumption
\biggl(
k2
k22
\biggr)
\in A2
2
[\ast ]
. Therefore,
\biggl(
k2
k22
\biggr)
\in A[\ast ] = A. Hence,
\biggl(
k2
k22
\biggr)
\in
\in A2
2. This proves (ii).
(iii) Because, A2
2 is self-adjoint and single-valued it holds
\{ 0\} = A2
2(0) = D
\Bigl(
A2
2
[\ast ]
\Bigr) [\bot ]
.
Hence, D
\Bigl(
A2
2
[\ast ]
\Bigr)
= D(A2
2) is dense in \scrK 2. Then also D
\bigl(
A1
2
\bigr)
= E2
\bigl(
D(A)
\bigr)
is dense in \scrK 2.
(iv) Because A is self-adjoint and A1
1 \subseteq A, the following implications hold:
A1
1 \subseteq A \Rightarrow A \subseteq A1
1
[\ast ] \Rightarrow D
\Bigl(
A1
1
[\ast ]
\Bigr) [\bot ]
\subseteq D(A)[\bot ].
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REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 901
It also holds A1
1(0) \subseteq D
\Bigl(
A1
1
[\ast ]
\Bigr) [\bot ]
, see the proof of (2.1). Because, A = A[\ast ] is closed we can
apply formula (2.1) to A. We get
A1
1(0) \subseteq D
\Bigl(
A1
1
[\ast ]
\Bigr) [\bot ]
\subseteq D(A)[\bot ] = A(0).
According to the assumption A(\scrK 1 \cap D(A)) \subseteq \scrK 1, it holds A(0) = A1
1(0) and, therefore, the \subseteq
signs become = signs in the above line, which proves (iv).
(v) A1
2
[\ast ]
(0) = D
\Bigl(
A1
2
[\ast ][\ast ]
\Bigr) [\bot ]
= D
\bigl(
A1
2
\bigr) [\bot ]
. According to (iii) D
\bigl(
A1
2
\bigr)
= D(A2
2) is dense in \scrK 2.
Therefore, A1
2
[\ast ]
(0) = \{ 0\} , which proves (v).
(vi) Let us first prove
A1
1 = A1
1
[\ast ] \leftrightarrow R
\bigl(
A1
2
\bigr)
\subseteq A(0).
(\Rightarrow ) Let us assume that A1
1
[\ast ]
= A1
1, and observe two arbitrary elements\Biggl(
h1 + h2
h11 + h12 + h22
\Biggr)
\in A,
\Biggl(
k1 + k2
k11 + k12 + k22
\Biggr)
\in A.
Because A is self-adjoint, it holds
[h1 + h2, k
1
1 + k12 + k22] = [h11 + h12 + h22, k1 + k2] \leftrightarrow
\leftrightarrow [h1, k
1
1 + k12] + [h2, k
2
2] = [h11 + h12, k1] + [h22, k2].
Because,A1
1 and A2
2 are symmetric this equation reduces to
[h1, k
1
2] = [h12, k1].
Because of A
\bigl(
D(A) \cap \scrK 1
\bigr)
\subseteq \scrK 1, we have h21 \equiv 0. Then, according to claim A1
2 \subseteq A2
1
[\ast ] in
Lemma 2.2, it holds
0 =
\bigl[
h21, k2
\bigr]
=
\bigl[
h1, k
1
2
\bigr]
= [h12, k1].
Hence, R
\bigl(
A1
2
\bigr)
\subseteq D(A1
1)
[\bot ]
= A1
1
[\ast ]
(0) = A1
1(0) = A(0).
(\Leftarrow =) Assume now that R
\bigl(
A1
2
\bigr)
\subseteq A(0) and prove that A1
1 is self-adjoint.
Assume that
\biggl(
k1
k11
\biggr)
\in A1
1
[\ast ]
. We will first prove that, for every
\Biggl(
h1 + h2
h11 + h12 + h22
\Biggr)
\in A,
it holds \bigl[
h1 + h2, k
1
1
\bigr]
=
\bigl[
h11 + h12 + h22, k1
\bigr]
. (2.2)
This equation is equivalent to \bigl[
h1, k
1
1
\bigr]
=
\bigl[
h11 + h12, k1
\bigr]
.
According to our assumption
\biggl(
k1
k11
\biggr)
\in A1
1
[\ast ]
, it holds
\bigl[
h1, k
1
1
\bigr]
=
\bigl[
h11, k1
\bigr]
.
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902 M. BOROGOVAC
It remains to prove 0 = [h12, k1].
According to our assumption, and (iv), it holds R
\bigl(
A1
2
\bigr)
\subseteq A(0) = A1
1(0) = D
\Bigl(
A1
1
[\ast ]
\Bigr) [\bot ]
. Then
we have
R
\bigl(
A1
2
\bigr)
[\bot ]D
\Bigl(
A1
1
[\ast ]
\Bigr)
\Rightarrow
\bigl[
h12, k1
\bigr]
= 0.
Hence, (2.2) is satisfied. It further means
\biggl(
k1
k11
\biggr)
\in A[\ast ] = A \Rightarrow
\biggl(
k1
k11
\biggr)
\in A1
1. This proves that A1
1 is
self-adjoint relation, i.e., it proves (\Leftarrow =).
Now (vi) follows from A(0) = A1
1(0).
(vii) Assume that A1
1 = A1
1
[\ast ]
. According to (i), we have
A = A1
1
\^+
\bigl(
A1
2 +A2
2
\bigr)
.
According to (vi), we obtain
A1
1 = A1
1
[\ast ] \leftrightarrow R(A1
2) \subseteq A1
1(0) \leftrightarrow
\leftrightarrow
\Biggl(
h2
h12 + h22
\Biggr)
=
\Biggl(
0
h12
\Biggr)
+
\Biggl(
h2
h22
\Biggr)
\forall
\Biggl(
h2
h12 + h22
\Biggr)
\in A1
2 +A2
2.
Therefore, for arbitrarily selected element from A = A1
1
\^+
\bigl(
A1
2 +A2
2
\bigr)
, it holds\biggl(
h1
h11
\biggr)
+
\biggl(
h2
h12 + h22
\biggr)
=
\biggl(
h1
h11
\biggr)
+
\biggl(
0
h12
\biggr)
+
\biggl(
h2
h22
\biggr)
.
From A1
2(0) \subseteq A1
1(0) it follows \biggl(
h1
h11
\biggr)
+
\biggl(
0
h12
\biggr)
\in A1
1.
Therefore, A = A1
1
\^+A2
2. Because of [h1, h2] = 0 and [h11+h12, h
2
2] = 0, we conclude A = A1
1 [+]A2
2.
Conversely, from
A = A1
1 [+]A2
2
and from Lemma 2.1 it follows that relations Ai
i are self-adjoint in the corresponding \scrK i, i = 1, 2.
(viii) This statement also follows from (2.1).
Theorem 2.1 is proved.
3. Direct sum representation of generalized Nevanlinna functions. 3.1. Let us assume that
functions Qi \in N\kappa i(\scrH ) are minimally represented by triplets (\scrK i, Ai,\Gamma i), i = 1, 2, in representations
of the form (1.1), where Ai are self-adjoint relations in Pontryagin spaces \scrK i and \Gamma i : \scrH \rightarrow \scrK i are
operators. We define the domain of Q := Q1 +Q2 by
\scrD (Q) = \scrD (Q1) \cap \scrD (Q2),
space \~\scrK as the orthogonal direct sum,
\~\scrK := \scrK 1 [+]\scrK 2 = \mathrm{c}.\mathrm{l}.\mathrm{s}.
\Biggl\{ \Biggl(
\Gamma 1z1h1
\Gamma 2z2h2
\Biggr)
: zi \in \scrD (Qi), hi \in \scrH , i = 1, 2
\Biggr\}
. (3.1)
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REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 903
Scalar product in \~\scrK is naturally defined by\Biggl[ \Biggl(
f1
f2
\Biggr)
,
\Biggl(
g1
g2
\Biggr) \Biggr]
:= [f1, g1] + [f2, g2], fi, gi \in \scrK i, i = 1, 2.
In this subsection we will create a minimal state space of Q within \~\scrK by means of the elements
\~\Gamma zh :=
\Biggl(
\Gamma 1zh
\Gamma 2zh
\Biggr)
, z \in \scrD (Q), h \in \scrH .
First, we will find state manifold of L(Q). We start with linear space
L := \mathrm{l}.\mathrm{s}.
\Bigl\{
\~\Gamma zh : z \in \scrD (Q), h \in \scrH
\Bigr\}
\subseteq \~\scrK . (3.2)
The closure of L in \~\scrK is given by
\=L = L[\bot ][\bot ]
= c.l.s.
\Bigl\{
\~\Gamma zh : z \in \scrD (Q), h \in \scrH
\Bigr\}
,
where L[\bot ] denotes the orthogonal complement of L in ( \~\scrK , [., .]). It is important to note that, in
general case, an indefinite scalar product [., .] may degenerate on the closure of a manifold even if
it does not degenerate on the given manifold (see [11, p. 39]). Later, we will prove that it is not the
case with L and \=L (see Lemma 4.1).
We define operator \~\Gamma =
\biggl(
\Gamma 1
\Gamma 2
\biggr)
: \scrH \rightarrow \scrK 1 [+]\scrK 2 by
\~\Gamma h := \Gamma 1h [+] \Gamma 2h, \Gamma ih \in \scrK i, i = 1, 2.
It holds \Bigl[
\~\Gamma h, k1 [+] k2
\Bigr]
=
\bigl(
h,\Gamma +
1 k1 + \Gamma +
2 k2
\bigr)
\forall k1 [+] k2 \in \~\scrK .
Therefore, \~\Gamma + : \scrK 1 [+]\scrK 2 \rightarrow \scrH satisfies
\~\Gamma + = \Gamma
+
1 + \Gamma +
2 ,
where we consider that \Gamma +
l , l = 1, 2, is extended on the whole space \scrK 1 [+]\scrK 2 by \Gamma +
i (kj) = 0
\forall kj \in \scrK j , j \not = i.
Let the functions Qi again be minimally represented by (1.1). For the function Q := Q1 +Q2,
consider the following representation:
Q(z) = Q1(z0)
\ast +Q2(z0)
\ast +
+(z - \=z0)(\Gamma
+
1 \Gamma
+
2 )
\Biggl(
I1 + (z - z0)(A1 - z) - 1 0
0 I2 + (z - z0)(A2 - z) - 1
\Biggr) \Biggl(
\Gamma 1
\Gamma 2
\Biggr)
, (3.3)
where z \in \scrD (Q) and Ii denote identities in \scrK i. Note that (3.3) is defined only when \Gamma 1 and \Gamma 2
simultaneously map the same vector h \in \scrH into \~\scrK . That means that manifold L is the linear span
of the vectors
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904 M. BOROGOVAC
\~\Gamma zh =
\Bigl(
I + (z - z0)( \~A - z) - 1
\Bigr)
\~\Gamma h, z \in \scrD (Q), h \in \scrH , (3.4)
where the resolvent is defined by
( \~A - z) - 1 :=
\Biggl(
(A1 - z) - 1 0
0 (A2 - z) - 1
\Biggr)
.
We know that the following holds:\biggl(
Qi(z) - Qi( \=w)
z - \=w
hz, hw
\biggr)
=
\bigl[
\Gamma izhz,\Gamma iwhw
\bigr]
, z, w \in \scrD (Qi), z \not = \=w; hz, hw \in \scrH ,
\bigl(
Qi
\prime (z)hz, h\=z
\bigr)
=
\bigl[
\Gamma izhz,\Gamma i\=zh\=z
\bigr]
, i = 1, 2.
Then it is easy to verify that for function Q = Q1 +Q2, the following holds:\biggl(
Q(z) - Q( \=w)
z - \=w
hz, hw
\biggr)
=
\bigl[
\~\Gamma zhz, \~\Gamma whw
\bigr]
, z, w \in \scrD (Q), z \not = \=w; hz, hw \in \scrH ,
\bigl(
Q\prime (z)hz, h\=z
\bigr)
=
\bigl[
\~\Gamma zhz, \~\Gamma \=zh\=z
\bigr]
.
According to thse equations we can, as in [7], identify building blocks of the state manifold L(Q)
with the building blocks of L \subseteq \~\scrK defined by (3.2). In other words, the following holds:
\varepsilon zh = \~\Gamma zh =
\Bigl(
I + (z - z0)
\bigl(
\~A - z
\bigr) - 1
\Bigr)
\~\Gamma h
and L = L(Q).
3.2. In Section 2, we have proved that relation A can be reduced in the sense of Definition 2.1
if it satisfies conditions of Theorem 2.1. In the following theorem we will describe decomposition
of Q in terms of the reducing nontrivial subspaces \scrK i and reducing relations Ai, i = 1, 2, of the
representing relation A of Q.
Theorem 3.1. (i) Assume:
(a) A function Q \in N\kappa (\scrH ) is minimally represented by (1.1) and there exist nondegenerate,
nontrivial subspaces \scrK 1 and \scrK 2 that reduce the representing relation A, i.e., A = A1 [+]A2. Then:
(b) \exists Qi\in N\kappa i(\scrH ), i = 1, 2, minimally represented by the triplets
\bigl(
\scrK i, Ai,\Gamma i
\bigr)
,
(c) Q(z) = Q1(z) +Q2(z), i = 1, 2,
(d) the representation (3.3) of Q is minimal, i.e., \scrK 1 [+]\scrK 2 is the minimal state space of Q.
(ii) Conversely, if conditions (b), (c) and (d) are satisfied, then the representation (3.3) is of the
form (1.1), and subspaces \scrK 1, \scrK 2 are reducing subspaces of \~A := A1 [+]A2, i.e., (a) holds.
(iii) In that case it holds \kappa 1 + \kappa 2 = \kappa .
Proof. (i) We know that negative index of the minimal state space \scrK is equal to \kappa , the negative
index of Q. Let \scrK 1 and \scrK 2 be nontrivial nondegenerate subspaces that reduce representing relation
A. Then \scrK = \scrK 1 [+]\scrK 2 and A = A1 [+]A2. If \kappa i, 0 \leq \kappa i, denote negative indexes of \scrK i, i = 1, 2,
then obviously \kappa 1 + \kappa 2 = \kappa .
Because A is a self-adjoint relation, according to Lemma 2.1, Ai are also self-adjoint relations
in \scrK i. Let Ei : \scrK \rightarrow \scrK i be orthogonal projections and \Gamma i := Ei \circ \Gamma , i = 1, 2. Then the following
decompositions hold:
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REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 905
I + (z - z0)(A - zI) - 1 =
=
\Biggl(
I1 + (z - z0)(A1 - z) - 1 0
0 I2 + (z - z0)(A2 - z) - 1
\Biggr)
(3.5)
and
Q(z) = Q1(z) +Q2(z),
where
Qi(z) := Qi(z0)
\ast =
\bigl(
z - \=z0
\bigr)
\Gamma +
i
\bigl(
I + (z - z0)(Ai - zI) - 1\bigr) \Gamma i. (3.6)
The constant operators Qi(z0)
\ast can be arbitrarily selected as long as Q1(z0)
\ast +Q2(z0)
\ast = Q(z0)
\ast .
Hence, the minimal representation (1.1) of Q can be expressed as the orthogonal sum representa-
tion (3.3). This proves (c) and (d).
Because Ai are self-adjoint linear relations in the Pontryagin spaces \scrK i, functions (3.6) are gener-
alized Nevanlinna functions. From (3.5) and from the minimality of representation (1.1), minimality
of representations (3.6) follows.
Indeed, for y1 [+] y2 \in \scrK 1 [+]\scrK 2 minimality of (1.1) means\Biggl[ \Biggl(
y1
y2
\Biggr)
,
\Biggl(
I1 + (z - z0(A1 - z) - 1 0
0 I2 + (z - z0)(A1 - z) - 1
\Biggr) \Biggl(
\Gamma 1h
\Gamma 2h
\Biggr) \Biggr]
= 0
\forall z \in \rho (A) \forall h \in \scrH \Rightarrow
\biggl(
y1
y2
\biggr)
= 0.
If we keep y2 = 0, we can conclude that Q1 is minimally represented by (\scrK 1, A1,\Gamma 1). By the
same token we can conclude that Q2 is minimally represented by (\scrK 2, A2,\Gamma 2). This further means
that negative indexes of functions Qi are equal to \kappa i, the negative indexes of space \scrK i. Hence,
Qi \in N\kappa i(\scrH ), i = 1, 2. This proves (b).
From the equation \kappa 1 + \kappa 2 = \kappa established for negative indexes of \scrK i and \scrK , now we can con-
clude that the same equation holds for negative indexes of the functions Qi and Q. This proves (iii).
(ii) Assume now that conditions (b), (c) and (d) are satisfied, where \~A := A1 [+]A2 is the
representing relation of Q. Then subspaces \scrK i and relations Ai satisfy conditions of Definition 2.1,
i.e., they are reducing subspaces and reducing relations of the representing relation \~A in (3.3).
Because, Ai are self-adjoint relations, according to Lemma 2.1 the relation \~A is also self-adjoint.
According to assumption (d) and Theorem 1.1, the triplet ( \~\scrK , \~A, \~\Gamma ) is uniquely determined (up to
isomorphism). Hence, representation (3.3) is of the form (1.1). This proves statement (a), which
completes the proof of (ii).
Theorem 3.1 is proved.
If the conditions of Theorem 2.1 are satisfied, then A2 is densely defined (single-valued) self-
adjoint operator in \scrK 2. In that case function Q2 has some nice features at infinity, see, e.g., [13]
(Satz 1.4) for scalar functions. If A2 is bounded, see [6] (Corollary 1) for operator valued functions.
3.3. As discussed in Subsection 3.1, vectors \~\Gamma zh, z \in \scrD (Q), h \in \scrH , are building blocks of
the state manifold L(Q) = L of Q := Q1 +Q2. Let us now consider the structure of \~\scrK introduced
by (3.1). Denote
L0 := \=L \cap L[\bot ].
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906 M. BOROGOVAC
Recall that the minimal state space \scrK of Q is defined as completion of the quotient space L\big/
L0
0
,
where
L0
0 := L \cap L[\bot ]
(see Section 1.1 or [9, 12] for more details).
Note, L0
0 \subseteq L0 in general case. We will see in Lemma 4.1 that in our setting it holds
L0
0 = \{ 0\} \Rightarrow L0 = \{ 0\} .
For our purpose, we need to decompose \~\scrK by means of L0. Obviously, L0 is finite-dimensional
because it is isotropic subspace of \=L \subseteq \~\scrK . According to [11] (Theorems 3.3 and 3.4) the following
decompositions hold:
\=L = L1 [+]L0, L[\bot ] = L0 [+]L2,
\~\scrK = L1 [+] (L0 \.+F )[+]L2, (3.7)
where L1 and L2 are non-degenerate subspaces and F is a neutral subspace of \~\scrK , skewly linked
to L0. Then \~\kappa 0 := \mathrm{d}\mathrm{i}\mathrm{m}L0 is the negative index of the non-degenerate subspace L0 \.+F. Let \~\kappa i,
i = 1, 2, denote the negative indexes of subspaces Li in decomposition (3.7). (\scrK , A,\Gamma ) again
denotes the triplet that minimaly represents Q = Q1 +Q2.
Theorem 3.2. Let functions Qi\in N\kappa i(\scrH ) be minimally represented by formulas of the form (1.1).
Assume that the function Q := Q1 + Q2\in N\kappa (\scrH ) is represented by orthogonal sum representa-
tion (3.3).
Then the subspace L1 in decomposition (3.7) is unitarily equivalent to the minimal state space
\scrK of the function Q = Q1+Q2. Therefore, \scrK and L1, including the corresponding scalar products,
can be identified, i.e., \scrK = L1 and \~\kappa 1 = \kappa .
Proof. Observe representation (3.3) of Q
Q(z) := Q(z0)
\ast + (z - \=z0)\~\Gamma
+
\Bigl(
I + (z - z0)
\bigl(
\~A - zI
\bigr) - 1
\Bigr)
\~\Gamma
and decomposition (3.7) of \~\scrK . In Subsection 3.1, we have proved that we can consider \varepsilon z = \~\Gamma z, i.e.,
we can identify manifold L defined by (3.2) with the state manifold L(Q), the starting manifold in
the building of the minimal state space \scrK of the given function Q. Therefore, we can use the usual
construction to obtain the minimal Pontryagin state space \scrK of Q by means of \~\Gamma z and L. Then we
will prove that \scrK is unitarily equivalent to L1.
Let us first prove that the minimal space \scrK of Q = Q1 + Q2, which is equal to completion
of L/L0
0
, is also equal to the completion of L/L0
. For that purpose, let us prove that the naturally
defined mapping
f + L0 \rightarrow f + L0
0 \forall f \in L (3.8)
is an isometric bijection between L\big/
L0
0
and L/L0
.
It obviously holds L0
0 \subseteq L0. Now we have
0 \not = f + L0 \in L/L0
\Rightarrow f /\in L0
0 \Rightarrow 0 \not = f + L0
0 \in L\big/
L0
0
.
In order to prove the converse implication, let us assume the contrary
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REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 907
0 \not = f + L0
0 \in L\big/
L0
0
and 0 = f + L0 \in L/L0
.
Then 0 = f + L0 means f \in L and f \in L0. It further means that f \in L and f [\bot ] \=L. Because,
\=L \supseteq L it follows f [\bot ]L; hence f \in L0
0, which is a contradiction. This proves that the naturally
defined mapping (3.8) is a bijection, and we can identify L\big/
L0
0
and L/L0
.
Recall that the scalar product is introduced in \scrK in the following manner: If for f, g, . . . \in L the
corresponding classes in the quotient space L\big/
L0
0
are denoted by \^f, \^g, . . . , then the scalar product is
defined by
\langle \^f, \^g\rangle := [f, g]. (3.9)
Then the quotient space L/L0
= L\big/
L0
0
can be completed in the usual way (see, e.g., [5], Section 2.4).
The completion \scrK is unitarily equivalent to space L1 introduced by equations (3.7).
Indeed, according to the above definition (3.9) and [11] (Theorem 2.4 (i)), the sequence \{ fn\} \subseteq
\subseteq L1 \cap L converges to some f0 \in L1 if and only if the sequence \{ \^fn\} \infty n=1 = \{ fn + L0\} \infty n=1 \subseteq \scrK
converges to \^f0 \in \scrK . Therefore, equation (3.9) extends to L1 and \scrK . This proves that \scrK and L1 are
unitarily equivalent and we can consider
\scrK = L1
and \~\kappa 1 = \kappa .
Theorem 3.2 is proved.
Remark 3.1. From (3.4) it follows
\~\Gamma z0 = \~\Gamma
and, therefore,
L0 \subset L[\bot ] \subseteq \mathrm{k}\mathrm{e}\mathrm{r} \~\Gamma +.
Hence, the operator \Gamma + : \scrK \rightarrow \scrH defined by \Gamma + \^f := \~\Gamma +f is well defined. If we also set \Gamma h := \~\Gamma h
\forall h \in \scrH , A := \~A| L1
in representation (3.3) of Q, then we obtain representation (1.1).
Corollary 3.1. Let functions Qi\in N\kappa i(\scrH ) be minimally represented by formulas of the form (1.1)
and Q := Q1 +Q2.The following statements hold:
(i) \~\scrK is the minimal state space of Q if and only if L1 = \=L = \~\scrK ; in that case \kappa = \kappa 1 + \kappa 2;
(ii) \kappa = \kappa 1 + \kappa 2 if and only if \~\scrK = L1 [+]L2, where L2 = \{ 0\} or L2 = L[\bot ] is a positive
subspace;
(iii) L0 = \{ 0\} is necessary but not sufficient condition for \kappa = \kappa 1 + \kappa 2.
Proof. (i) Assume, \~\scrK is minimal state space of Q. According to first equation of (3.7) it holds
L1 \subseteq \=L \subseteq \~\scrK . According to Theorem 3.2, L1 is minimal state space of Q. Therefore, L1 = \=L = \~\scrK .
Conversely, if L1 = \=L = \~\scrK holds, then minimality of \~\scrK follows from Theorem 3.2. Then
\kappa = \kappa 1 + \kappa 2 follows from Theorem 3.1.
(ii) Assume \kappa = \kappa 1 + \kappa 2. That means that the numbers of negative squares of L1 and \~\scrK are
equal, and \~\kappa = 0. According to (3.7) it must be L0 = \{ 0\} . Therefore, \~\scrK = L1 [+]L2, where
L2 = \{ 0\} or L2 is a positive subspace.
Conversely, \~\scrK = L1 [+]L2 and L2 = \{ 0\} or L2 is positive, means that the numbers of negative
squares of L1 and \~\scrK are equal, i.e., \kappa = \kappa 1 + \kappa 2.
In Example 4.2 we will prove that there exists the case where \~\scrK = L1 [+]L2 and L2 is positive.
(iii) \kappa = \kappa 1 + \kappa 2 \Rightarrow L0 = \{ 0\} \Rightarrow L0
0 = \{ 0\} . In Example 4.3 we will show that there exists the
case where \~\scrK = L1 [+]L2 and L2 is negative subspace. That is an example where it holds L0 = \{ 0\}
and \kappa < \kappa 1 + \kappa 2.
Corollary 3.1 is proved.
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908 M. BOROGOVAC
4. Analytic criteria. 4.1. In this section, we will prove criteria that enable us to research the
underlying state space, and negative index of the sum Q := Q1 +Q2 analytically, without knowing
operator representations of Q, Q1, and Q2. In order to derive equations in those criteria we will have
to use Definition 1.1 and definitions of scalar products in terms of formal sums (see Subsection 1.1).
Let us consider any function Q \in N\kappa (\scrH ). By definition \kappa is the maximal (finite) number of
negative squares of the sesquilinear form [., .] defined by the sums
n\sum
i,j=1
\bigl[
\Gamma zihi,\Gamma zjhj
\bigr]
:=
n\sum
i,j=1
\biggl(
Q(zi) - Q( \=zj)
zi - \=zj
hi, hj
\biggr)
, (4.1)
where zl \in \scrD (Q), hl \in \scrH , l = 1, . . . , n. In other words, \kappa is the negative index of the state manifold\bigl(
L(Q), [., .]
\bigr)
. According to Theorem 1.1, the negative index of the minimal state space \scrK is also
equal to \kappa .
Let us now focus on the sum Q = Q1 +Q2. Then sum (4.1) can be written as
n\sum
i,j=1
\biggl( \biggl(
Q1(zi) - Q1( \=zj)
zi - \=zj
+
Q2(zi) - Q2( \=zj)
zi - \=zj
\biggr)
hi, hj
\biggr)
=
n\sum
i,j=1
\Biggl[ \Biggl(
\Gamma 1zihi
\Gamma 2zihi
\Biggr)
,
\Biggl(
\Gamma 1zjhj
\Gamma 2zjhj
\Biggr) \Biggr]
,
where zl \in \scrD (Q1) \cap \scrD (Q2) =: \scrD (Q), hl \in \scrH , l = 1, . . . , n. Such sums are subset of sums (4.2)
below, which generate the inner product in \~\scrK := \scrK 1 [+]\scrK 2. Indeed, here Q1 and Q2 take the same
domain points zl \in \scrD (Q), while in (4.2) Q1 and Q2 take domain points zl \in \scrD (Q1) and \zeta l \in \scrD (Q2)
independently. This means that the space \~\scrK created by means of the sums (4.2) may be larger than
the state space \scrK , which is created by means of the sums (4.1).
Now we can prove the following lemma.
Lemma 4.1. Assume that functions Qi\in N\kappa i(\scrH ) are minimally represented by triplets (\scrK i, Ai,
\Gamma i), i = 1, 2, and Q := Q1 +Q2. If scalar product does not degenerate on the state manifold L =
= L(Q), i.e., if L0
0 = \{ 0\} , then scalar product does not degenerate on \=L, and it holds \~\scrK = \scrK [+]L2,
where \scrK = \=L is the minimal state space of Q.
Proof. According to (3.2), L \subseteq \~\scrK . Let us assume that form [., .] induced by (4.1) in the state
manifold L = L(Q) does not degenerate, i.e., L0
0 = \{ 0\} . Then L\big/
L0
0
= L, and the minimal state
space \scrK is by definition equal to the completion of L.
Because Pontryagin space \~\scrK is complete, the closure \=L \subseteq \~\scrK is also complete. Then it holds
L \subseteq \scrK \subseteq \=L.
Pontryagin space, \scrK is nondegenerate. Because, completion \scrK is a closed set in \~\scrK , and \=L is the
smallest closed set which contains L, we conclude \scrK = \=L. Hence, \=L is nondegenerate.
Then according to (3.7) it holds \~\scrK = \scrK [+]L2.
Lemma 4.1 is proved.
4.2. By definition of \~\scrK (see (3.1)), the negative index \kappa := \kappa 1+\kappa 2 of \~\scrK is equal to the maximal
number of negative squares of the form defined by means of the sums
n\sum
i,j=1
\Biggl[ \Biggl(
\Gamma 1zihi
\Gamma 2\zeta ifi
\Biggr)
,
\Biggl(
\Gamma 1zjhj
\Gamma 2\zeta jfj
\Biggr) \Biggr]
=
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REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 909
=
n\sum
i,j=1
\biggl(
Q1(zi) - Q1( \=zj)
zi - \=zj
hi, hj
\biggr)
+
\biggl(
Q2(\zeta i) - Q2( \=\zeta j)
\zeta i - \=\zeta j
fi, fj
\biggr)
, (4.2)
where zl \in \scrD (Q1), \zeta l \in \scrD (Q2); hl, fl \in \scrH , l = 1, . . . , n. Because points zl, \zeta l are arbitrarily
selected in their domains, we can create the following sums out of (4.2):
n\sum
i,j=1
\Biggl[ \Biggl(
\Gamma 1wihi
\Gamma 2wihi
\Biggr)
,
\Biggl(
\Gamma 1wjhj
\Gamma 2wjhj
\Biggr) \Biggr]
+
n\sum
i,j=1
\Biggl[ \Biggl(
\Gamma 1zihi
\Gamma 2\zeta ifi
\Biggr)
,
\Biggl(
\Gamma 1zjhj
\Gamma 2\zeta jfj
\Biggr) \Biggr]
, (4.3)
where wl \in \scrD (Q), zl \in \scrD (Q1), \zeta l \in \scrD (Q2), and the second sum is created by vectors that satisfy
condition \Biggl(
\Gamma 1zlhl
\Gamma 2\zeta lfl
\Biggr)
[\bot ]L.
Note that the first sum here is associated with L. The orthogonality condition for vectors from the
second sum in (4.3) can be written with simplified notation as\Biggl[ \Biggl(
\Gamma 1zh1
\Gamma 2\zeta h2
\Biggr)
,
\Biggl(
\Gamma 1wg
\Gamma 2wg
\Biggr) \Biggr]
= 0 \forall w \in \scrD (Q) \forall g \in \scrH ,
where z \in \scrD (Q1), \zeta \in \scrD (Q2), hi \in \scrH , i = 1, 2. Because, scalar product (., .) in \scrH is nondegene-
rate, this condition can be written as the equation
Q1(z) - Q1( \=w)
z - \=w
h1 +
Q2(\zeta ) - Q2( \=w)
\zeta - \=w
h2 = 0 \forall w \in \scrD (Q). (4.4)
Lemma 4.2. Let Q \in N\kappa (\scrH ) be any minimally represented function by a triplet (\scrK , A,\Gamma ).
(i) If there exist z \in \scrD (Q) such that \mathrm{k}\mathrm{e}\mathrm{r} \Gamma z \not = \{ 0\} , then
\mathrm{k}\mathrm{e}\mathrm{r} \Gamma z = \mathrm{k}\mathrm{e}\mathrm{r} \Gamma w =: \mathrm{k}\mathrm{e}\mathrm{r} \Gamma \forall w \in \scrD (Q).
(ii) h \in \mathrm{k}\mathrm{e}\mathrm{r} \Gamma if and only if
Q(z) - Q( \=w)
z - \=w
h = 0 \forall z \forall w \in \scrD (Q).
Proof. (i) For function Q minimally represented by (1.1), it holds \rho (A) = \scrD (Q) and
\Gamma z =
\bigl(
I + (z - w)(A - z) - 1
\bigr)
\Gamma w \forall w \in \rho (A) = \scrD (Q)
(see [9, 10]). Assume the contrary to the claim (i), that for some w \in \scrD (A) it holds \Gamma wh \not = 0,
\Gamma zh = 0. Then we have
0 = \Gamma zh =
\bigl(
I + (z - w)(A - z) - 1
\bigr)
\Gamma wh \Rightarrow (z - w)(A - z) - 1\Gamma wh = - \Gamma wh.
According to [1] (2.11) it holds
(A - z)(A - z) - 1 \supseteq I \Rightarrow (z - w)\Gamma wh \subseteq - (A - z)\Gamma wh.
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910 M. BOROGOVAC
Therefore, w\Gamma wh \in A(\Gamma wh), i.e., w is an eigenvalues of A. This contradicts to the fact that w is a
regular point of A. This proves \mathrm{k}\mathrm{e}\mathrm{r} \Gamma z \subseteq \mathrm{k}\mathrm{e}\mathrm{r} \Gamma w. The converse inclusion is obvious. This proves the
first equation of (i).
Because, \mathrm{k}\mathrm{e}\mathrm{r} \Gamma w is independent of w \in \scrD (Q), we can introduce \mathrm{k}\mathrm{e}\mathrm{r} \Gamma := \mathrm{k}\mathrm{e}\mathrm{r} \Gamma w, w \in \scrD (Q). It
is obvious now that claim (i) holds for any two points z, w \in \scrD (Q). This completes the proof of (i).
(ii) We have
h \in \mathrm{k}\mathrm{e}\mathrm{r} \Gamma \leftrightarrow [\Gamma zh,\Gamma wg] = 0 \forall z \in \scrD (Q) \forall w \in \scrD (Q) \forall g \in \scrH \leftrightarrow
\leftrightarrow \Gamma +
w\Gamma zh = 0 \forall z \in \scrD (Q) \forall w \in \scrD (Q) \leftrightarrow Q(z) - Q( \=w)
z - \=w
h = 0 \forall z \forall w \in \scrD (Q).
The following statement is a criteria that identifies zero-symbols \varepsilon zh = \Gamma zh, i.e., the symbols
that do not play any role in the state manifold L(Q).
Corollary 4.1. Let Q \in N\kappa (\scrH ) be a minimally represented function by a triplet (\scrK , A,\Gamma ). If
there exists a solution (z0, h) \in \scrD (Q)\times \scrH , h \not = 0, of the equation
Q(z) - Q( \=w)
z - \=w
h = 0 \forall w \in \scrD (Q), (4.5)
then \Gamma zh = 0 \forall z \in \scrD (Q).
It is easy to find regular matrix functions that satisfy (4.5), i.e., that have \mathrm{k}\mathrm{e}\mathrm{r} \Gamma \not = \{ 0\} .
Example 4.1. Consider the following regular matrix functions:
Q(z) =
\Biggl(
z + a z
z z + b
\Biggr)
\in N\kappa
\bigl(
\bfC 2
\bigr)
; a, b \in R, (a, b) \not = (0, 0), \kappa \in \{ 0, 1, 2\} .
Then, for vector h =
\biggl(
1
- 1
\biggr)
, identity (4.5) holds.
Now we can classify solutions of equation (4.4). According to Lemma 4.2, if
\biggl(
h1
h2
\biggr)
\in \mathrm{k}\mathrm{e}\mathrm{r} \Gamma 1 \times
\times \mathrm{k}\mathrm{e}\mathrm{r} \Gamma 2, then for both functions Qi it holds
Qi(zi) - Qi( \=w)
zi - \=w
hi \equiv 0.
Let us call such solutions
\biggl(
h1
h2
\biggr)
of (4.4) singular solutions. Then, according to Lemma 4.2, the
vectors
\biggl(
\Gamma 1z1h1
\Gamma 2z2h2
\biggr)
\equiv 0 \forall
\biggl(
z1
z2
\biggr)
\in \scrD (Q1)\times \scrD (Q2), i = 1, 2, i.e., they do not exist in \~\scrK . Therefore,
we can exclude singular solutions of (4.4) from the following considerations about structure of
\~\scrK , without loss of generality. Hence, in the following definitions we assume that we deal only
with nonsingular solutions. It is consistent with the standard assumption that the functions \Gamma z are
injections.
The obvious solutions (z1, z2;h1, h2) \in \scrD (Q1) \times \scrD (Q2) \times \scrH \times \scrH of (4.4), i.e., the solutions
with h1 = h2 = 0, we call trivial solutions. Hence, the nonsingular solutions of (4.4) with
(h1, h2) \not = (0, 0), we call nontrivial. We will solve equation (4.4), later in couple of examples.
Let us introduce expression
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REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 911
E = E(z1, z2;h1, h2) :=
\biggl(
Q1(z1) - Q1( \=z1)
z1 - \=z1
h1, h1
\biggr)
+
\biggl(
Q2(z2) - Q2( \=z2)
z2 - \=z2
h2, h2
\biggr)
,
i.e.,
E = E(z1, z2;h1, h2) :=
\Biggl[ \Biggl(
\Gamma 1z1h1
\Gamma 2z2h2
\Biggr)
,
\Biggl(
\Gamma 1z1h1
\Gamma 2z2h2
\Biggr) \Biggr]
.
A nontrivial solution (z1, z2;h1, h2) of (4.4) we call positive, negative, neutral if it satisfies
E(z1, z2;h1, h2) > 0, < 0, = 0, respectively.
For z1 = z2 = z and h1 = h2 = h we get an important special case of equation (4.4):\biggl(
Q1(z) - Q1( \=w)
z - \=w
+
Q2(z) - Q2( \=w)
z - \=w
\biggr)
h = 0 \forall w \in \scrD (Q). (4.6)
Why is this equation important? Equation (4.6) identifies when term
Q1(z) - Q1( \=w)
z - \=w
cancels
out with term
Q2(z) - Q2( \=w)
z - \=w
. That is, how a negative square is lost, i.e., the negative index is
reduced in sum Q1 +Q2. Then in the underlying space \~\scrK we have the following:
Assume that (z;h) is a nontrivial (and nonsingular) solution of (4.6). That means that there
exists a nonzero vector \~\Gamma zh :=
\biggl(
\Gamma 1zh
\Gamma 2zh
\biggr)
\in \~K. On the other hand, according to Corollary 4.1, for the
symbol \Gamma zh corresponding to Q(z) in the minimal state space \scrK of Q it holds \Gamma zh = 0. Hence, we
learn that (0 \not =)\~\Gamma zh \in L0
0 \subset \~\scrK corresponds to (0 =)\Gamma zh := \~\Gamma zh+ L0
0 \in L\big/
L0
0
\subseteq \scrK .
Let us interpret this explanation in terms of almost Pontryagin spaces. Recall that an almost
Pontryagin space is a Pontryagin space to which a finite dimensional degenerate linear space has
been added orthogonaly (see [17]).
According to Theorem 3.2 we have \=L = L1 [+]L0 = \scrK [+]L0, i.e., \=L is an almost Pontryagin
space, with isotropic subspace L0. By similar method we obtain almost Pontryagin spaces \=Li =
= Ki [+]Li
0, i = 1, 2. Because Li
0 \cap \~K = \{ 0\} , i = 1, 2, the overlap \=L1 \cap \=L2 does not have any
nonzero elements in \~\scrK . The symbols \Gamma z, z \in \scrD (Q), h \in \scrH that belong to the overlap are char-
acterized as singular solutions of the equation (4.4), and excluded from the considerations. Hence,
the overlap does not affect the negative index \kappa . However, the negative index \kappa is affected by the
existence of nonzero elements \~\Gamma zh in the isotropic subspace L0 of the almost Pontryagin space \=L.
Those elements are characterized by the nonsingular, nontrivial solutions of the equation (4.6).
The following theorem gives us further analytic means to investigate structure of the state space
\~\scrK and to compare number of negative squares \kappa of Q := Q1 +Q2 with the sum \kappa 1 + \kappa 2.
Theorem 4.1. Assume Qi \in N\kappa i(\scrH ) are functions minimally represented by triplets (\scrK i, Ai,\Gamma i),
i = 1, 2, and Q := Q1 +Q2 is represented by ( \~\scrK , \~A,\~\Gamma ), i.e., by (3.3).
(i) There exists a nontrivial solution of equation (4.4) if and only if \~\scrK is not minimal state space
of Q.
(ii) Equation (4.6) has a nontrivial solution (z, h) \in \scrD (Q)\times \scrH if and only if \varepsilon zh := \~\Gamma zh \in L0
0.
(iii) If any nontrivial solution of equation (4.4) is neutral or negative, then \kappa < \kappa 1 + \kappa 2.
(iv) If all nontrivial solutions of (4.4) are positive, then L[\bot ] = L0 [+]L2 is a nonnegative
subspace of \~\scrK , where L2 is positive definite subspace. In this case \kappa = \kappa 1 + \kappa 2 if and only if the
state manifold L = L(Q) is nondegenerate.
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912 M. BOROGOVAC
(v) A necessary condition for \kappa = \kappa 1 +\kappa 2 is that equation (4.6) has only trivial solutions. Evan
the stronger condition, that equation (4.4) has only trivial solutions, is not sufficient for \kappa = \kappa 1+\kappa 2.
(Recall, we exclude singular solutions.)
Proof. According to Theorem 3.2 and Corollary 3.1, the following possibilities exist:
(a) \~\scrK = \=L=L1,
(b) \~\scrK = L1 [+]L2, where L2 is a nonpositive (nondegenerate) subspace,
(c) \~\scrK = L1 [+]L2, where L2 is a positive subspace,
(d) \~\scrK = L1 [+] (L0 \.+F )[+]L2.
(i) By defintion the existence of the nontrivial (which is also nonsingular) solution (z1, z2;h1, h2)
of (4.4) means that for at least one function Qi, i = 1, 2, it holds
hi \not = 0 \wedge \Gamma izihi \not = 0.
In other words, the existence of the nontrivial solution (z1, z2;h1, h2) of (4.4) is equivalent to\Biggl(
\Gamma 1z1h1
\Gamma 2z2h2
\Biggr)
\not =
\Biggl(
0
0
\Biggr)
and \biggl(
Q1(z1) - Q1( \=w)
z1 - \=w
h1, g
\biggr)
+
\biggl(
Q2(z2) - Q2( \=w)
z2 - \=w
h2, g
\biggr)
=
=
\Biggl[ \Biggl(
\Gamma 1z1h1
\Gamma 2z2h2
\Biggr)
,
\Biggl(
\Gamma 1wg
\Gamma 2wg
\Biggr) \Biggr]
= 0 \forall w \in \scrD (Q) \forall g \in \scrH .
This is equivalent to existence of
0 \not =
\Biggl(
\Gamma 1z1h1
\Gamma 2z2h2
\Biggr)
\in L[\bot ].
This is further equivalent to the claim that one of the cases (b), (c), or (d) is satisfied, which is
according to Corollary 3.1 (i) equivalent to the claim that \~\scrK is not minimal state space of Q.
(ii) In Subsection 3.2 we showed that we can identify \varepsilon zh = \~\Gamma z. Solution (z;h) is a nontrivial
solution of (4.6) if and only if \~\Gamma zh \in L and
\bigl[
\~\Gamma zh, \~\Gamma wg
\bigr]
= 0 \forall w \in \scrD (Q) \forall g \in \scrH . This is equivalent
to 0 \not = \~\Gamma zh \in L \cap L[\bot ], i.e., it is an isotropic element in L.
(iii) If nontrivial and nonpositive solutions of (4.4) exist, then (b) or (d) holds. Therefore,
\kappa = \~\kappa 1 < \~\kappa = \kappa 1 + \kappa 2.
(iv) Let us first prove the claim: if G = l.s.\{ x : [x, x] > 0\} , then G is a positive manifold.
If x and y are two positive and linearly dependent vectors, i.e., y = \beta x, \beta \not = - 1, then obviously
[x+ y, x+ y] > 0.
Assume now that x and y are two positive and linearly independent vectors. For every \alpha =
= | \alpha | ei\varphi \in C it holds | \alpha | 2[x, x] = [\alpha x, \alpha x] = | \alpha | 2[ei\varphi x, ei\varphi x] > 0. Because of this property, in
the sequel we can consider \alpha \in R in the linear combinations of the form \alpha x + y, without loss of
generality.
Then, for every \alpha \in R and two positive independent vectors x, y \in G, it holds
P (\alpha ) :=
\bigl[
\alpha x+ y, \alpha x+ y
\bigr]
= \alpha 2[x, x] + 2Re[x, y]\alpha + [y, y] \geq 0.
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REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 913
Quadratic polynomial P (\alpha ) \geq 0 because [x, x] > 0 and its discriminant is nonpositive, according
to Cauchy – Schwartz inequality. As we know, equality sign in Cauchy – Schwartz inequality holds
only when x and y are two linearly dependent vectors. Hence, for positive independent vectors
x, y \in G, which we have here, it holds [\alpha x+ y, \alpha x+ y] > 0. Because we already proved that linear
combination of two linearly dependent positive vectors x and y is positive, we can claim that linear
combination of any two positive vectors is a positive vector.
Then the positivity of a linear combination of n positive vectors follows by induction.
Assume now that all solutions of (4.4) are positive. According to the above claim, quadratic form
in the second sum of (4.3) is positive. Then the scalar product in the subspace
L[\bot ] = \mathrm{c}.\mathrm{l}.\mathrm{s}.
\Biggl\{ \Biggl(
\Gamma 1z1h1
\Gamma 2z2h2
\Biggr)
: zi \in \scrD (Qi),
\Biggl[ \Biggl(
\Gamma 1z1h1
\Gamma 2z2h2
\Biggr)
, \~\Gamma wg
\Biggr]
= 0 \forall w \in \scrD (Q) \forall g \in \scrH
\Biggr\}
is nonnegative or positive definite. We know that L[\bot ] = L0 [+]L2, where L0 is isotropic subspace
of L[\bot ] and L2 is a positive definite subspace (see [11], Theorem 3.3). Therefore, if there exist a
neutral vector e \in L[\bot ], then it has to be in L0. That is equivalent to \~\kappa 1 < \kappa 1+\kappa 2. L0 \not = \{ 0\} means
that \=L is degenerate. According to Lemma 4.1, then L(Q) is also degenerate.
If L0 = \{ 0\} we have case (c), which is equivalent to \kappa = \kappa 1+\kappa 2. In Example 4.2 we will prove
existence of the case (c).
(v) If we assume, in contrast to the first claim of (v), that (z;h) is a nontrivial solution of (4.6),
then we get \Biggl[ \Biggl(
\Gamma 1zh
\Gamma 2zh
\Biggr)
,
\Biggl(
\Gamma 1wg
\Gamma 2wg
\Biggr) \Biggr]
= 0 \forall w \in \scrD (Q) \forall g \in \scrH .
This means that 0 \not = \~\Gamma zh \in L0
0. According to Corollary 3.1 (iii), it holds \kappa < \kappa 1 + \kappa 2. This is a
contradiction that proves the first claim of (v).
In Example 4.3 we will see that even when equation (4.4) has only trivial solution it is possible
to have \kappa < \kappa 1 + \kappa 2. That will prove the second claim of (v).
Theorem 4.1 is proved.
The following theorem gives us some analytic tools to research existence of positive, negative,
isotropic and neutral vectors in \=L.
Theorem 4.2. Assume that Qi\in N\kappa i(\scrH ) are minimally represented by triplets (\scrK i, Ai,\Gamma i), i =
= 1, 2, and Q := Q1 +Q2 is represented by ( \~\scrK , \~A,\~\Gamma ).
(i) There exists
e := \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\~\Gamma znhzn
\bigl(
\in \=L
\bigr)
, e \not = 0, (4.7)
if and only if it holds:
(a)
\biggl(
Q1(zn) - Q1( \=w1)
zn - \=w1
hzn , g
1
\biggr)
+
\biggl(
Q2(zn) - Q2( \=w2)
zn - \=w2
hzn , g
2
\biggr)
\rightarrow a(w1, w2, g
1, g2) \not \equiv 0 (n \rightarrow
\rightarrow \infty ) \forall wi \in \scrD (Qi) \forall gi \in \scrH , i = 1, 2,
(b)
\biggl( \biggl(
Q1(zn) - Q1( \=zn)
zn - \=zn
+
Q2(zn) - Q2( \=zn)
zn - \=zn
\biggr)
hzn , hzn
\biggr)
\rightarrow b \not = \mp \infty (n \rightarrow \infty ) for some
sequences \{ zn\} \infty n=1 \subseteq \scrD (Q), \{ hn\} \infty n=1 \subseteq \scrH .
In that case e \in \=L is positive, neutral, negative element if and only if b >, =, < 0, respectively.
(ii) If in addition to (a) and (b) it holds
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914 M. BOROGOVAC
(c)
\biggl( \biggl(
Q1(zn) - Q1( \=w)
zn - \=w
+
Q2(zn) - Q2( \=w)
zn - \=w
\biggr)
hzn , g
\biggr)
\rightarrow 0 (n \rightarrow \infty ) w \in \scrD (Q) \forall g \in \scrH ,
then element e given by (4.7) is an isotropic vector of \=L, b = 0, \~\scrK is not minimal state space of Q
and \kappa < \kappa 1 + \kappa 2.
Proof. (i) Let us assume that (4.7) holds. According to [11] (Theorem 2.4), it is equivalent to\Biggl[
\~\Gamma znhzn ,
\Biggl(
\Gamma 1w1g
1
\Gamma 2w2g
2
\Biggr) \Biggr]
\rightarrow
\Biggl[
e,
\Biggl(
\Gamma 1w1g
1
\Gamma 2w2g
2
\Biggr) \Biggr]
\not \equiv 0 (n \rightarrow \infty ), wi \in \scrD (Qi), gi \in \scrH , i = 1, 2,
and \bigl[
\~\Gamma znhzn , \~\Gamma znhzn
\bigr]
\rightarrow [e, e](n \rightarrow \infty ).
Those limits can be written as\biggl(
Q1(zn) - Q1( \=w1)
zn - \=w1
hzn , g
1
\biggr)
+
\biggl(
Q2(zn) - Q2( \=w2)
zn - \=w2
hzn , g
2
\biggr)
\rightarrow
\rightarrow
\Biggl[
e,
\Biggl(
\Gamma 1w1g
1
\Gamma 2w2g
2
\Biggr) \Biggr]
=: a \not \equiv 0 (n \rightarrow \infty )
and \biggl( \biggl(
Q1(zn) - Q1( \=zn)
zn - \=zn
+
Q2(zn) - Q2( \=zn)
zn - \=zn
\biggr)
hzn , hzn
\biggr)
\rightarrow [e, e] =: b (n \rightarrow \infty ).
Because, b := [e, e], the last statement of (i) holds by definition. This proves (i).
(ii) Assume now that condition (c) is satisfied as well. Then for all w \in \scrD (Q) and g \in \scrH it
holds \biggl( \biggl(
Q1(zn) - Q1( \=w)
zn - \=w
+
Q2(zn) - Q2( \=w)
zn - \=w
\biggr)
hzn , g
\biggr)
=
\Bigl[
\~\Gamma znhzn , \~\Gamma wg
\Bigr]
\rightarrow
\bigl[
e, \~\Gamma wg
\bigr]
= 0,
when n \rightarrow \infty . This means that e \not = 0, and e \in \=L \cap L[\bot ] = L0. Hence, it must be b = 0. According
to Corollary 3.1 (iii) it must be \kappa < \kappa 1 + \kappa 2.
Theorem 4.2 is proved.
4.3. The following simple examples clarify the previous statements. In addition, they serve as
proofs of existence of the cases theoretically anticipated in Corollary 3.1 and Theorem 4.1.
Example 4.2. Consider the following matrix functions that satisfy conditions of Theorem 4.1:
Q1(z) = -
\Biggl(
z - 1 0
0 z - 1
\Biggr)
\in N0
\bigl(
\bfC 2
\bigr)
, Q2(z) = -
\Biggl(
z - 2 z - 1
z - 1 0
\Biggr)
\in N1
\bigl(
\bfC 2
\bigr)
.
Then
Q(z) := Q1(z) +Q2(z) = -
\Biggl(
z - 1 + z - 2 z - 1
z - 1 z - 1
\Biggr)
.
We can then solve (4.4):
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REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 915
\left(
1
z1 \=w
0
0
1
z1 \=w
\right) \Biggl( h11
h12
\Biggr)
+
\left(
z2 + \=w
z22 \=w
2
1
z2 \=w
1
z2 \=w
0
\right)
\Biggl(
h21
h22
\Biggr)
= 0 \forall w \in \scrD (Q).
Solving this system gives
h1 =
\Biggl(
h11
0
\Biggr)
, h2 =
\left( 0
- z2
z1
h11
\right) .
Then we easily verify that all nontrivial solutions are positive. Indeed,
E =
\Biggl(
Q1 (z1) - Q1( \=z1)
z1 - \=z1
\Biggl(
h11
0
\Biggr)
,
\Biggl(
h11
0
\Biggr) \Biggr)
+
\left( Q2 (z2) - Q2 ( \=z2)
z2 - \=z2
\left( 0
- z2
z1
h11
\right) ,
\left( 0
- z2
z1
h11
\right) \right) =
= 2
\bigm| \bigm| h11\bigm| \bigm| 2
| z1| 2
> 0.
According to Theorem 4.1 (i), the representation (3.3) is not minimal.
To determine whether the number of negative squares is preserved we can apply Definition 1.1.
We can take n = 1, z1 \in C+, \mathrm{R}\mathrm{e} z1 < 0, h =
\biggl(
1
- 1
\biggr)
. Then from
\bigl(
NQ(z1, z1)h1, h1
\bigr)
=
2\mathrm{R}\mathrm{e}(z1)
| z1| 4
< 0
and \kappa 1 + \kappa 2 = 1, we conclude \kappa = 1. Hence, we have that all nontrivial solutions are positive, and
number of negative squares is preserved even though representation (3.3) is not minimal. This also
proves existence of the case (c) in the proof of Theorem 4.1.
We have already proved that L[\bot ] contains positive elements. Because of \kappa = \kappa 1 + \kappa 2 = 1
we know that \=L is nondegenerate. Therefore, L2 := L[\bot ] is a positive subspace. This proves the
existence of the case anticipated in Corollary 3.1 (ii) and thus completes the proof of Corollary 3.1 (ii).
Note that without Theorem 4.1, we would have to find operator representations of the functions
Qi and Q to obtain the above answers, which would make the task much more difficult.
Example 4.3. Consider the functions Q1(z) := - 2z - 1 - z - 2 \in N1 and Q2(z) := 2z - 1 \in N1.
Then
Q(z) := Q1(z) +Q2(z) = - z - 2 \in N1.
Hence, \kappa 1 + \kappa 2 = 2 > 1 = \kappa .
In this example, (4.4) is given by\biggl(
2
z1 \=w
+
z1 + \=w
z21 \=w
2
\biggr)
h1 -
2
z2 \=w
h2 = 0 \forall w \in \scrD (Q).
This equation has only the trivial solution h1 = h2 = 0. Hence, this is an example of the sum
Q := Q1 + Q2 that has only a trivial solution of (4.4) and still does not preserve the number of
negative squares. This completes the proof of Theorem 4.1 (v).
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916 M. BOROGOVAC
According to Theorem 4.1 the subspace L[\bot ] should be non-positive. We will prove that L[\bot ] is
negative. This will also prove existence of the case (b) in the proof of Theorem 4.1. In order to do
that we will use operator representations:
Q1(z) := \Gamma +
1 (A1 - zI) - 1\Gamma 1 = - 2z - 1 - z - 2 \in N1,
where
A1 =
\Biggl(
0 1
0 0
\Biggr)
, J1 =
\Biggl(
0 1
1 0
\Biggr)
, \Gamma 1 =
\Biggl(
1
1
\Biggr)
, \Gamma +
1 = \Gamma \ast
1J1 =
\Biggl(
1
1
\Biggr) \Biggl(
0 1
1 0
\Biggr)
and
Q2(z) := \Gamma +
2 (A2 - zI) - 1\Gamma 2 = 2z - 1 \in N1,
where
A2 = (0), J2 = ( - 1), \Gamma 2 =
\Bigl(
2
\surd
2
\Bigr)
, \Gamma +
2 = \Gamma \ast
2J2 = -
\Bigl(
2
\surd
2
\Bigr)
.
According to the definitions in Section 3, we have
\~A =
\left(
0 1 0
0 0 0
0 0 0
\right) , \~J =
\left(
0 1 0
1 0 0
0 0 - 1
\right) , \~\Gamma =
\left(
1
1
\surd
2
\right) ,
\~\Gamma + =
\bigl(
1 1
\surd
2
\bigr) \left(
0 1 0
1 0 0
0 0 - 1
\right) ,
\~\Gamma z =
\bigl(
\~A - zI
\bigr) - 1\~\Gamma h = -
\left(
z - 1 z - 2 0
0 z - 1 0
0 0 z - 1
\right)
\left(
1
1
\surd
2
\right) h = -
\left(
z - 1 + z - 2
z - 1
\surd
2z - 1
\right) h \in L,
and
\=L = \mathrm{c}.\mathrm{l}.\mathrm{s}.
\left\{ -
\left(
z - 1 + z - 2
z - 1
\surd
2z - 1
\right) h : z \in \scrD (Q), h \in \scrH
\right\} .
Then, for y =
\left( y1
y2
y3
\right) \in L[\bot ]\subseteq \~\scrK = \scrK 1 [+]\scrK 2, we obtain
0 =
\Bigl[
y,
\bigl(
\~A - zI
\bigr) - 1\~\Gamma h
\Bigr]
=
\left(
\left(
y1
y2
y3
\right) , \~J
\left(
- z - 1 - z - 2
- z - 1
-
\surd
2z - 1
\right) h
\right) =
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REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 917
=
\left(
\left(
y1
y2
y3
\right) ,
\left(
- z - 1
- z - 1 - z - 2
\surd
2z - 1
\right) h
\right) \forall z \in \scrD (Q) \forall h \in \scrH \Rightarrow
\Rightarrow y =
\left(
y1
0
y1
\big/ \surd
2
\right) ,
[y, y] =
\left(
\left(
y1
0
y1
\big/ \surd
2
\right) , \~J
\left(
y1
0
y1
\big/ \surd
2
\right)
\right) = - | y1| 2
2
< 0.
Hence, vector \bfy \in L[\bot ] is strictly negative. This is indeed the case (b) anticipated in the proof of
Theorem 4.1.
5. The final decomposition of \bfitQ . 5.1. Let the function Q \in N\kappa (\scrH ) be minimally represented
by (1.1) and \alpha \in R be a generalized pole of Q that is not of positive type. It is customary to say
that A and \Gamma are closely connected if representation (1.1) is minimal. Let us decompose the function
Q by means of the Jordan chains of the representing relation A at \alpha .
According to [6] (Lemma 1), there is no loss of generality to assume that \alpha \in R is a single
generalized pole that is not of positive type. In that case A is an operator. For given eigenvector x0
of A at \alpha \in R, let us denote by X one of the maximal Jordan chains of x0. Let us denote by
S\alpha (x0) := \mathrm{l}.\mathrm{s}.\{ X\} .
Let the Hilbert subspace, denoted here by \scrK 0 \subset \scrK , consist of all positive eigenvectors of the
representing operator A at \alpha . Let E0 : \scrK \rightarrow \scrK 0 be the orthogonal projection E\prime := I - E0, \scrK \prime :=
:= E\prime \scrK and \Gamma 0 := E0\Gamma . Subspaces \scrK 0 and \scrK \prime obviously reduce operator A. We define \Gamma \prime := E\prime \Gamma
and A\prime := E\prime AE\prime .
Now let x10, . . . , x
1
l1 - 1 be a maximal nondegenerate Jordan chain of A\prime at \alpha in the Pontryagin
space \scrK \prime . We define the projection: E1 : \scrK \prime \rightarrow S\alpha (x
1
0), and subspace \scrK 1 := E1\scrK \prime . Then A1 =
= E1A
\prime E1 and \Gamma 1 := E1\Gamma
\prime are closely connected operators. Let \kappa 1 denote the negative index of the
Pontryagin space \scrK 1.
We can repeat these steps until we exhaust all nondegenerate Jordan chains. At every step we
can decompose the corresponding function as in Theorem 3.1.
Assume that there are r > 0 such (nondegenerate) chains at \alpha . We introduce E := E0+E1+ . . .
. . .+Er. Then \scrK = E\scrK [+] (I - E)\scrK . Let us introduce Er+1 := I - E, \scrK r+1 := Er+1\scrK , \Gamma r+1 =
= Er+1\Gamma . Subspaces E\scrK and \scrK r+1 obviously reduce A. From the construction of the Pontryagin
space \scrK r+1 we conclude that all degenerate chains of A at \alpha are in \scrK r+1.
By using the above notation, we can summarize these results in the following proposition.
Proposition 5.1. Let \alpha \in R be a generalized pole that is not of positive type of Q \in N\kappa (\scrH ),
where Q is given by minimal representation (1.1). Then
\scrK = \scrK 0 [+]\scrK 1 [+] . . . [+]\scrK r [+]\scrK r+1, (5.1)
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918 M. BOROGOVAC
where r \in \bfN is the number of independent nondegenerate Jordan chains of A at \alpha ;\scrK i are A-
invariant Pontryagin subspaces of indices \kappa i, i = 0, 1, . . . , r, r + 1, respectively; \kappa 0 = 0, \kappa =
=
\sum r+1
i=1
\kappa i. For every i = 1, 2, . . . , r, subspace \scrK i is a linear span of the corresponding maximal
nondegenerate Jordan chain xi0, . . . , x
i
li - 1. All positive eigenvectors are in \scrK 0. All degenerate chains
of A at \alpha are in \scrK r+1.
The corresponding nontrivial decomposition Q := Q0 + Q1 + . . . + Qr + Qr+1 satisfies \kappa =
=
\sum r+1
i=0
\kappa i.
5.2. Because \scrK i, i = 1, . . . , r, is a linear span of a maximal Jordan chain, it does not have
a nontrivial invariant subspaces of A. In Proposition 5.1, we separated non-degenerate maximal
Jordan chains Xi and Xj by A-invariant disjoint subspaces \scrK i and \scrK j , i.e., Xi \subset \scrK i, X
j \subset \scrK j ,
\scrK i \cap \scrK j = \{ 0\} \forall i \not = j. The following natural question arises: Is it possible to separate degenerate
Jordan chains in a similar way? More precisely:
Let A be a self-adjoint operator in a Pontryagin space \scrK . Given two degenerate maximal Jordan
chains Xi, i = 1, 2, at an eigenvalue \alpha \in R, is it possible to find an A-invariant nondegenerate
subspace \scrK 1 such that it holds X1 \subset \scrK 1 and X2 \cap \scrK 1 = \varnothing ?
In order to address this question, we introduce the following model with two independent degen-
erate chains at \alpha = 0 of the first order, i.e., two neutral eigenvectors. We denote \langle k\rangle = l.s.\{ k\} .
Proposition 5.2. Assume that
\scrK = \scrH [+]
\bigl( \bigl(
\langle x10\rangle [+]\langle x20\rangle
\bigr)
\dotplus (\langle f1\rangle [+]\langle f2\rangle )
\bigr)
, (5.2)
J =
\left(
I 0 0 0 0
0 0 0 1 0
0 0 0 0 1
0 1 0 0 0
0 0 1 0 0
\right)
, A =
\left(
A11 0 0 a1 a2
(., a1) 0 0 \alpha 1 0
(., a2) 0 0 0 \alpha 2
0 0 0 0 0
0 0 0 0 0
\right)
,
where
\bigl(
\scrH , (., .)
\bigr)
is a Hilbert space, A11 is a bounded self-adjoint operator on \scrH , 0 \not = \alpha i \in R,
0\not =ai \in \scrH , i = 1, 2, are linearly independent. Then:
(i) operator A is a self-adjoint operator in the Pontryagin space \scrK ;
(ii) vectors xi0 are neutral, simple eigenvectors of A at \alpha = 0 and f i = Jxi0, i = 1, 2;
(iii) if operator A11: \scrH \rightarrow \scrH is irreducible, then operator A does not have any eigenvalues
different from \alpha = 0;
(iv) if operator A11 is irreducible, then operator A does not have any invariant nondegenerate
subspace that contains one eigenvector xi0 and not the other, xj0, i \not = j, i, j = 1, 2.
Proof. For vectors from \scrK we will use notation (h, \beta 1, \beta 2, \gamma 1, \gamma 2)
T , h \in \scrH , \gamma i, \beta i \in C, i = 1, 2.
Statements (i) and (ii) are straightforward verification.
(iii) In contrast to the statement, assume that the operator A has the eigenvalue \beta \not = 0. Then
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REDUCIBILITY OF SELF-ADJOINT LINEAR RELATIONS AND APPLICATION TO GENERALIZED . . . 919
A
\left(
h
\beta 1
\beta 2
\gamma 1
\gamma 2
\right)
= \beta
\left(
h
\beta 1
\beta 2
\gamma 1
\gamma 2
\right)
\Rightarrow
\left(
A11h+ a1\gamma 1 + a2\gamma 2
(h, a1) + \alpha 1\gamma 1
(h, a2) + \alpha 2\gamma 2
0
0
\right)
= \beta
\left(
h
\beta 1
\beta 2
\gamma 1
\gamma 2
\right)
.
Hence, \gamma i = 0, i = 1, 2. Therefore,
A
\left(
h
\beta 1
\beta 2
0
0
\right)
=
\left(
A11h
(h, a1)
(h, a2)
0
0
\right)
= \beta
\left(
h
\beta 1
\beta 2
0
0
\right)
\Rightarrow h \not = 0 and A11h = \beta h.
This means that operator A11 has a nonzero eigenvalue \beta . Because A11 is a bounded self-adjoint
operator on \scrH , it is reduced by the eigenvector h \in \scrH , see also definition of reductibility in [2]
(Section 40). That contradicts the assumption that A11 is irreducible. This proves (iii).
(iv) In contrast to the statement, assume that operator A11 is irreducible in \scrH and that there exist
a nondegenerate, A-invariant, nontrivial subspace \scrK 1 of \scrK such that x10 \in \scrK 1 and x20 /\in \scrK 1. Then
\scrK 1 must contain f1 ; otherwise, according to (5.2), the subspace \scrK 1 would be degenerate. Similarly,
\scrK 1 cannot contain f2, because then \scrK 1 without x20 would be degenerate. Hence, vectors from \scrK 1
must satisfy \gamma 1 \not = 0,\gamma 2 = 0 , and \scrH 1 := \scrH \cap \scrK 1 must contain vectors of the form A11h+a1\gamma 1 \in \scrH .
This means that \scrK 1 is of the form
\scrK 1 = \scrH 1 [+]
\bigl(
\langle x10\rangle \dotplus \langle f1\rangle
\bigr)
, (5.3)
where \scrH 1 \not = \{ 0\} . It is easy to verify, that it holds x20 \in \scrK [\bot ]
1 .
For an arbitrarily selected k1 \in \scrK 1, we have
k1 :=
\left(
h
\beta 1
0
\gamma 1
0
\right)
\in \scrK 1 \Rightarrow A
\left(
h
\beta 1
0
\gamma 1
0
\right)
=
\left(
A11h+ a1\gamma 1
(h, a1) + \alpha 1\gamma 1
(h, a2)
0
0
\right)
\in \scrK 1, \beta 1, \gamma 1 \in C, h \in \scrH 1.
Because \scrK 1 is A-invariant, Ak1 must be of the form (5.3). Hence, it must be
(h, a2) = 0 \forall h \in \scrH 1
and
A11h+ a1\gamma 1 \in \scrH 1 \forall h \in \scrH 1 \forall \gamma 1 \in \bfC .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
920 M. BOROGOVAC
Hence a2\bot \scrH 1, where 0 \not = a2 \in \scrH . This means \{ 0\} \subsetneq \scrH 1 \subsetneq \scrH , i.e., \scrH 1 is a nontrivial subspace
of \scrH .
If we set \gamma 1 = 0 in the second equation, then we conclude that A11h \in \scrH 1 \forall h \in \scrH 1. Therefore,
\scrH 1 is an A11-invariant, nontrivial subspace in \scrH . Because A11 is bounded self-adjoint operator on
the Hilbert space \scrH , operator A11 is reduced by \scrH 1 (see again [2], Section 40). That contradicts the
assumption of irreducibility of A11 and proves (iv).
This example shows that there does not exist an A-invariant subspace that contains one and not
the other degenerate eigenvector of A at \alpha .
Corollary 5.1. There is no nontrivial decomposition of \scrK r+1 and Qr+1, i.e., decomposition (5.1)
of \scrK and corresponding decomposition of Q are final.
References
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\kappa , Oper. Theory Adv. and Appl., 28, 17 – 26 (1988).
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Received 24.04.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
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| id | umjimathkievua-article-6084 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:03Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c7/fd41a66691f63e4dbd1bf52585ae0fc7.pdf |
| spelling | umjimathkievua-article-60842022-10-24T09:23:11Z Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions Borogovac, M. Borogovac, M. Generalized Nevanlinna function; Linear relation; Operator representation; Jordan chain UDC 517.9We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function $Q$ represented by a self-adjoint linear relation $A$ in a Pontryagin space is decomposed by means of the reducing subspaces of $A.$ The sum of two functions $Q_{i}{\in N}_{\kappa_{i}}(\mathcal{H}),$ $i=1, 2,$ minimally represented by the triplets $(\mathcal{K}_{i},A_{i},\Gamma_{i})$ is also studied. For this purpose, we create a model $(\tilde{\mathcal{K}},\tilde{A},\tilde{\Gamma })$ to represent $Q:=Q_{1}+Q_{2}$ in terms of $(\mathcal{K}_{i},A_{i},\Gamma_{i})$. By using this model, necessary and sufficient conditions for $\kappa =\kappa_{1}+\kappa_{2}$ are proved in the analytic form. Finally, we explain how degenerate Jordan chains of the representing relation $A$ affect the reducing subspaces of $A$ and the decomposition of the corresponding function $Q.$ УДК 517.9 Звiднiсть самоспряжених лiнiйних спiввiдношень i застосування до узагальнених функцiй Неванлiнни Наведено необхiднi та достатнi умови звiдностi самоспряженого лiнiйного спiввiдношення у просторi Крейна. Далi узагальнена функцiя Неванлiнни $Q$, що представлена самоспряженим лiнiйним спiввiдношенням $A$ у просторi Понтрягiна, розкладається за допомогою звiдних пiдпросторiв $A$. Також вивчається сума двох функцiй $Q_{i}{\in N}_{\kappa_{i}}(\mathcal{H}),$ $i=1, 2,$ мiнiмально представлена трiйками $(\mathcal{K}_{i},A_{i},\Gamma_{i})$. З цiєю метою створено модель $(\tilde{\mathcal{K}},\tilde{A},\tilde{\Gamma })$, що представляє $Q:=Q_{1}+Q_{2}$ в термiнах $(\mathcal{K}_{i},A_{i},\Gamma_{i})$. За допомогою цiєї моделi необхiднi та достатнi умови для $\kappa =\kappa_{1}+\kappa_{2}$ доведено в аналiтичнiй формi. Насамкiнець ми пояснюємо, яким чином виродженi жордановi ланцюги представницьких спiввiдношень $A$ впливають на звiднi пiдпростори $A$ та на розклад вiдповiдної функцiї $Q.$ Institute of Mathematics, NAS of Ukraine 2022-08-09 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6084 10.37863/umzh.v74i7.6084 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 7 (2022); 893 - 920 Український математичний журнал; Том 74 № 7 (2022); 893 - 920 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6084/9274 Copyright (c) 2022 Muhamed Borogovac |
| spellingShingle | Borogovac, M. Borogovac, M. Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions |
| title | Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions |
| title_alt | Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions |
| title_full | Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions |
| title_fullStr | Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions |
| title_full_unstemmed | Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions |
| title_short | Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions |
| title_sort | reducibility of self-adjoint linear relations and application to generalized nevanlinna functions |
| topic_facet | Generalized Nevanlinna function Linear relation Operator representation Jordan chain |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6084 |
| work_keys_str_mv | AT borogovacm reducibilityofselfadjointlinearrelationsandapplicationtogeneralizednevanlinnafunctions AT borogovacm reducibilityofselfadjointlinearrelationsandapplicationtogeneralizednevanlinnafunctions |