Isolated eigenvalues, poles and compact perturbations of Banach space operators

Abstract

Given a Banach space operator A, the isolated eigenvalues E(A) and the poles Π(A) (resp., eigenvalues E a (A) which are isolated points of the approximate point spectrum and the left ploles Π a (A)) of the spectrum of A satisfy Π(A) ⊆ E(A) (resp., Π a (A) ⊆ 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 Π(A) ⊆ E a (A), but no such inclusion exists for E(A) and Π a (A). The study of identities E(A) = Π a (A) and E a (A) = Π(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. (P 1) : E(A) = Π a (A) and (P 2) : E a (A) = Π(A)and their stability under perturbations by commuting Riesz operators, have been studied in a number of papers in the recent past, amongst them [2,3,4,8,9,17,18,20,21].