Perturbation determinants in Banach spaces - with an application to eigenvalue estimates for perturbed operators

TL;DR

This paper introduces perturbation determinants in Banach spaces and applies them to derive new bounds on eigenvalues of perturbed operators and generators of $C_0$-semigroups, extending recent results.

Contribution

It provides a self-contained introduction to perturbation determinants in Banach spaces and uses them to establish novel eigenvalue bounds for perturbed operators and semigroup generators.

Findings

New bounds on discrete eigenvalues of compactly perturbed operators

Extended eigenvalue estimates for generators of $C_0$-semigroups

Broader applicability of perturbation determinants in Banach spaces

Abstract

In the first part of this paper we provide a self-contained introduction to (regularized) perturbation determinants for operators in Banach spaces. In the second part, we use these determinants to derive new bounds on the discrete eigenvalues of compactly perturbed operators, broadly extending some recent results by Demuth et al. In addition, we also establish new bounds on the discrete eigenvalues of generators of $C_0$-semigroups.