Desirable Decompositions of Generalized Nevanlinna Functions

TL;DR

This paper investigates specific decompositions of generalized Nevanlinna functions, providing new operator representations and conditions for such decompositions, which enhance understanding of their structure and singularities.

Contribution

It introduces new sufficient conditions for desirable decompositions of generalized Nevanlinna functions and presents a novel operator representation of the inverse function in Pontryagin spaces.

Findings

Provided sufficient conditions for desirable decompositions.

Developed a new operator representation of the inverse function.

Gained insights into the singularities of the inverse function.

Abstract

For a given generalized Nevanlinna function $Q\in N_{\kappa }\left( H \right)$, we study decompositions that satisfy: $Q=Q_{1}+Q_{2}$; $Q_{i}{\in N}_{\kappa_{i}}\left( H \right)$, and $\kappa_{1}+\kappa_{2}=\kappa $, $0\le \kappa_{i}$, which we call desirable decompositions. In this paper, some sufficient conditions for such decompositions of $Q$ are given. One of the main results is a new operator representation of $\hat{Q}\left(z\right):=-{Q(z)}^{-1}$ if $Q\left( z \right):=\Gamma_{0}^{+}\left( A-z\right)^{-1}\Gamma_{0}$, where $A$ is a bounded self-adjoint operator in a Pontryagin space. The new representation is used to get an interesting desirable decomposition of $\hat{Q}$ and to obtain some information about singularities of $\hat{Q}$.