Estimating the number of eigenvalues of linear operators on Banach spaces

TL;DR

This paper develops bounds on the number of eigenvalues of perturbed operators on Banach spaces, generalizing classical results and extending previous work from Hilbert spaces using complex analysis and a novel finite-dimensional approach.

Contribution

It introduces a new method for estimating eigenvalues of operators on Banach spaces without relying on determinant theory, broadening the scope of spectral analysis.

Findings

Bounds on eigenvalues depend on approximation numbers of the perturbation

Generalizes classical eigenvalue distribution results to Banach spaces

Provides examples showing differences from Hilbert space cases

Abstract

Let $L_0$ be a bounded operator on a Banach space, and consider a perturbation $L=L_0+K$, where $K$ is compact. This work is concerned with obtaining bounds on the number of eigenvalues of $L$ in subsets of the complement of the essential spectrum of $L_0$, in terms of the approximation numbers of the perturbing operator $K$. Our results can be considered as wide generalizations of classical results on the distribution of eigenvalues of compact operators, which correspond to the case $L_0=0$. They also extend previous results on operators in Hilbert space. Our method employs complex analysis and a new finite-dimensional reduction, allowing us to avoid using the existing theory of determinants in Banach spaces, which would require strong restrictions on $K$. Several open questions regarding the sharpness of our results are raised, and an example is constructed showing that there are some essential differences in the possible distribution of eigenvalues of operators in general Banach spaces, compared to the Hilbert space case.