Compact perturbations and consequent hereditarily polaroid operators

TL;DR

This paper investigates hereditarily polaroid operators on Banach and Hilbert spaces, establishing conditions under which such operators can be obtained via compact perturbations, and clarifying spectral properties related to SVEP.

Contribution

It provides necessary and sufficient conditions for operators to be hereditarily polaroid after compact perturbations, answering a question about spectral properties and SVEP.

Findings

Operators in ${ m{ extbf{HP}}}$ have SVEP on $\

A connected component $\

Existence of compact $K$ such that $A+K$ is hereditarily polaroid.

Abstract

A Banach space operator $A\in B({\cal{X}})$ is polaroid, $A\in {\cal{P}}$, if the isolated points of the spectrum $\sigma(A)$ are poles of the operator; $A$ is hereditarily polaroid, $A\in{\cal{HP}}$, if every restriction of $A$ to a closed invariant subspace is polaroid. Operators $A\in{\cal{HP}}$ have SVEP on $\Phi_{sf}(A)=\{\lambda: A-\lambda$ is semi Fredholm $\}$: This, in answer to a question posed by Li and Zhou (Studia Math. 221(2014), 175-192), proves the necessity of the condition $\Phi_{sf}^+(A)=\emptyset$. A sufficient condition for $A\in B({\cal{X}})$ to have SVEP on $\Phi_{sf}(A)$ is that its component $\Omega_a(A)=\{\lambda\in\Phi_{sf}(A): \rm{ind}(A-\lambda)\leq 0\}$ is connected. We prove: If $A\in B({\cal{H}})$ is a Hilbert space operator, then a necessary and sufficient condition for there to exist a compact operator $K$ such that $A+K\in{\cal{HP}}$ is that $\Omega_a(A)$ is connected.