Eigenvalues of compactly perturbed operators via entropy numbers

TL;DR

This paper provides improved estimates for the number of discrete eigenvalues of compactly perturbed operators on Banach spaces, using entropy number ideals and avoiding complex analysis tools.

Contribution

It introduces new bounds for eigenvalues of perturbed operators based on weak entropy number ideals, enhancing previous results without complex analysis.

Findings

Derived new eigenvalue estimates for compactly perturbed operators

Improved bounds over previous results by the author and others

Utilized Carl's inequality as the main proof tool

Abstract

We derive new estimates for the number of discrete eigenvalues of compactly perturbed operators on Banach spaces, assuming that the perturbing operator is an element of a weak entropy number ideal. Our results improve upon earlier results by the author and by Demuth et al. The main tool in our proofs is an inequality of Carl. In particular, in contrast to all previous results we do not rely on tools from complex analysis.