Compact perturbations and consequent hereditarily polaroid operators

Abstract

A Banach space operator $A\in B(\mathcal{X})$ is polaroid, $A\in \mathcal{P}$, if the isolated points of the spectrum $\sigma (A)$ are poles of the operator; $A$ is hereditarily polaroid, $A\in \mathcal{H}\mathcal{P}$, if every restriction of $A$ to a closed invariant subspace is polaroid. Operators $A\in \mathcal{H}\mathcal{P}$ 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 ${\Phi }_{sf}^{+}(A)=\varnothing$. A sufficient condition for $A\in B(\mathcal{X})$ to have SVEP on ${\Phi }_{sf}(A)$ is that its component ${\Omega }_a(A)=\{\lambda \in {\Phi }_{sf}(A):\mathrm{ind}(\mathrm{A}-\lambda )\le 0\}$ is connected. We prove: If $A\in B(\mathcal{H})$ is a Hilbert space operator, then a necessary and sufficient condition for there to exist a compact operator $K$ such that $A+K\in \mathcal{H}\mathcal{P}$ is that ${\Omega }_a(A)$ is connected.

arXiv comment10 Pages