Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions

DOI10.1007/S11253-022-02118-XarXiv2010.00725MaRDI QIDQ6350355

Author name not available (Why is that?)

Publication date: 1 October 2020

Abstract: Necessary and sufficient conditions for reducidibility of a self-adjoint linear relation in a Krein space are given. Then a generalized Nevanlinna function

Q

, represented by a self-adjoint linear relation

A

, is decomposed by means of the reducing subspaces of

A

. The sum of two functions

Q_{i} {i n N}_{k a p p a_{i}} l e f t (m a t h c a l H i g h t), h i n s p a c e i = 1, h i n s p a c e 2

, minimally represented by the triplets

l e f t (m a t h c a l K_{i}, A_{i}, G a m m a_{i} i g h t)

, is also studied. For that purpose, a model

(i l d e m a t h c a l K, i l d e A, i l d e G a m m a)

to represent

Q := Q_{1} + Q_{2}

in terms of

l e f t (m a t h c a l K_{i}, A_{i}, G a m m a_{i} i g h t)

is created. By means of that model, necessary and sufficient conditions for

k a p p a = k a p p a_{1} + k a p p a_{2}

are proven in analytic terms. At the end, it is explained how degenerate Jordan chains of the representing relation

A

affect reducing subspaces of

A

and decomposition of the corresponding function

Q

.

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