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

TL;DR

This paper establishes criteria for the reducibility of self-adjoint linear relations in Krein spaces and applies these results to decompose generalized Nevanlinna functions, providing new insights into their structure and representation.

Contribution

It introduces necessary and sufficient conditions for reducibility of self-adjoint linear relations and develops a model for decomposing generalized Nevanlinna functions based on these relations.

Findings

Criteria for reducibility of self-adjoint linear relations in Krein spaces.

A model for representing sums of generalized Nevanlinna functions.

Analysis of how Jordan chains influence reducing subspaces.

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 $Q_{i}{\in N}_{\kappa_{i}}\left( \mathcal{H} \right),\thinspace i=1,\thinspace 2$, minimally represented by the triplets $\left( \mathcal{K}_{i},A_{i},\Gamma_{i} \right)$, is also studied. For that purpose, a model $( \tilde{\mathcal{K}},\tilde{A},\tilde{\Gamma } )$ to represent $Q:=Q_{1}+Q_{2}$ in terms of $\left( \mathcal{K}_{i},A_{i},\Gamma_{i} \right)$ is created. By means of that model, necessary and sufficient conditions for $\kappa =\kappa_{1}+\kappa_{2}$ 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 $Q$.