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.
Categories: math.FA