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

M.   Borogovac

doi:10.37863/umzh.v74i7.6084

M. Borogovac Boston Mutual Life Insurance Company, Canton, MA, USA https://orcid.org/0000-0001-6857-8309

DOI: https://doi.org/10.37863/umzh.v74i7.6084

Keywords: Generalized Nevanlinna function; Linear relation; Operator representation; Jordan chain

Abstract

UDC 517.9
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 $Q_{i}{\in N}_{\kappa_{i}}(\mathcal{H}),$ $i=1, 2,$ 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 $Q.$

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