Compact perturbations resulting in hereditarily polaroid operators

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. It is seen that operators $A\in(\cal {HP})$ have SVEP - the single-valued extension property - on $\Phi_{sf}(A)=\{\lambda: A-\lambda$ is semi Fredholm $\}$. Hence $\Phi^+_{sf}(A)=\{\lambda\in\Phi_{sf}(A), \ind(A-\lambda)>0\}=\emptyset$ for operators $A\in(\cal {HP})$, and a necessary and sufficient condition for the perturbation $A+K$ of an operator $A\in B({\cal X})$ by a compact operator $K\inB({\cal X})$ to be hereditarily polaroid is that $\Phi_{sf}^+(A)=\emptyset$. A sufficient condition for $A\in B({\cal X})$ to have SVEP on $\Phi_{sf}(A)$ is that its component $\Omega_a(A)=\{\lambda\in\Phi_{sf}(A): \ind(A-\lambda)\leq 0\}$ 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 $K\in B({\cal H})$ such that $A+K\in(\cal {HP})$ is that $\Omega_a(A)$ is connected.