Isolated eigenvalues, poles and compact perturbations of Banach space operators

TL;DR

This paper investigates the relationships between isolated eigenvalues and poles of Banach space operators, focusing on their stability under compact perturbations and exploring specific operator classes like Toeplitz and shift operators.

Contribution

It extends the study of identities between eigenvalues and poles to include non-commuting compact perturbations, broadening previous results on commuting Riesz operators.

Findings

Identifies conditions for stability of eigenvalue-pole identities under compact perturbations.

Provides examples involving Toeplitz and shift operators to illustrate theoretical results.

Shows that certain spectral identities are preserved or fail under specific perturbations.

Abstract

Given a Banach space operator $A$, the isolated eigenvalues $E(A)$ and the poles $\Pi(A)$ (resp., eigenvalues $E^a(A)$ which are isolated points of the approximate point spectrum and the left ploles $\Pi^a(A)$) of the spectrum of $A$ satisfy $\Pi(A)\subseteq E(A)$ (resp., $\Pi^a(A)\subseteq E^a(A)$), and the reverse inclusion holds if and only if $E(A)$ (resp., $E^a(A)$) has empty intersection with the B-Weyl spectrum (resp., upper B-Weyl spectrum) of $A$. Evidently $\Pi(A)\subseteq E^a(A)$, but no such inclusion exists for $E(A)$ and $\Pi^a(A)$. The study of identities $E(A)=\Pi^a(A)$ and $E^a(A)=\Pi(A)$, and their stability under perturbation by commuting Riesz operators, has been of some interest in the recent past. This paper studies the stability of these identities under perturbation by (non-commuting) compact operators. Examples of analytic Toeplitz operators and operators satisfying the abstract shift condition are considered.