On the index of meromorphic operator-valued functions and some applications

TL;DR

This paper explores the algebraic multiplicities and index theory of meromorphic operator-valued functions, with applications to perturbation theory, Birman-Schwinger operators, and Weyl-Titchmarsh functions in Hilbert spaces.

Contribution

It introduces a unified approach to algebraic multiplicities and the index of meromorphic operator-valued functions, linking these concepts to various applications in operator theory.

Findings

Unified framework for algebraic multiplicities of zeros

Connections between index theory and perturbation analysis

Applications to Weyl-Titchmarsh functions and dual pairs

Abstract

We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces. Applications to abstract perturbation theory and associated Birman-Schwinger-type operators and to the operator-valued Weyl-Titchmarsh functions associated to closed extensions of dual pairs of closed operators are provided.