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

Muhamed Borogovac

TL;DR

The paper analyzes reducibility of self-adjoint linear relations in Krein spaces and applies these results to generalized Nevanlinna functions by decomposing Q via reducing subspaces of the representing relation A. It develops a direct-sum state-space framework to represent sums Q=Q_{1}+Q_{2}, establishing analytic criteria for when the negative index satisfies $\kappa=\kappa_{1}+\kappa_{2}$ and detailing how degeneracies and Jordan chains influence the decomposition. A minimal direct-sum model is constructed to realize sums of generalized Nevanlinna functions, linking the algebraic structure of A with index additivity and the minimality of representations. The work further provides a final Jordan-chain based decomposition of Q at real poles not of positive type, clarifying how positive, nondegenerate, and degenerate blocks contribute to the overall function and its spectral properties. Together, these results yield practical criteria for when decompositions preserve the index and how the underlying state-space geometry governs the analytic behavior of generalized Nevanlinna functions.

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

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

TL;DR

The paper analyzes reducibility of self-adjoint linear relations in Krein spaces and applies these results to generalized Nevanlinna functions by decomposing Q via reducing subspaces of the representing relation A. It develops a direct-sum state-space framework to represent sums Q=Q_{1}+Q_{2}, establishing analytic criteria for when the negative index satisfies and detailing how degeneracies and Jordan chains influence the decomposition. A minimal direct-sum model is constructed to realize sums of generalized Nevanlinna functions, linking the algebraic structure of A with index additivity and the minimality of representations. The work further provides a final Jordan-chain based decomposition of Q at real poles not of positive type, clarifying how positive, nondegenerate, and degenerate blocks contribute to the overall function and its spectral properties. Together, these results yield practical criteria for when decompositions preserve the index and how the underlying state-space geometry governs the analytic behavior of generalized Nevanlinna functions.

Abstract

Necessary and sufficient conditions for reducidibility of a self-adjoint linear relation in a Krein space are given. Then a generalized Nevanlinna function , represented by a self-adjoint linear relation , is decomposed by means of the reducing subspaces of . The sum of two functions , minimally represented by the triplets , is also studied. For that purpose, a model to represent in terms of is created. By means of that model, necessary and sufficient conditions for are proven in analytic terms. At the end, it is explained how degenerate Jordan chains of the representing relation affect reducing subspaces of and decomposition of the corresponding function .

Paper Structure

This paper contains 5 sections, 15 theorems, 162 equations.

Table of Contents

  1. Preliminaries and introduction
  2. Reducing subspaces of the self-adjoint linear relation in the Krein space
  3. Direct sum representation of generalized Nevanlinna functions
  4. Analytic criteria
  5. The final decomposition of Q

Key Result

Theorem 1.2

A function $Q:\mathcal{D}(Q)\to L(\mathcal{H})$ is a generalized Nevanlinna function of the index $\kappa$, denoted by $Q\in N_{\kappa}(\mathcal{H})$, if and only if it has a representation of the form where $A$ is a self-adjoint linear relation in some Pontryagin space $(\mathcal{K}, [.,.])$ of the index $\tilde{\kappa}\ge \kappa; \Gamma:\mathcal{H}\to \mathcal{K}$ is a bounded operator. (Obviou

Theorems & Definitions (21)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Corollary 3.4
  • ...and 11 more