Compact perturbations resulting in hereditarily polaroid operators

Abstract

A Banach space operator A ∈ B(X ) is polaroid, A ∈ (P), if the isolated points of the spectrum σ(A) are poles of the operator; A is hereditarily polaroid, A ∈ (HP), if every restriction of A to a closed invariant subspace is polaroid.This, in answer to a question posed by Li and Zhou [17, Problem 5.5], proves the necessity of the condition Φis a Hilbert space operator, then a necessary and sufficient condition for there to exist a compact operator K ∈ B(H) such that A + K ∈ (HP) is that Ω a (A) is connected.