Hereditarily polaroid operators, SVEP and Weyl’s theorem

A Banach space operator T ∈ B(X) is hereditarily polaroid, T ∈ HP, if every part of T is polaroid. HP operators have SVEP. It is proved that if T ∈ B(X) has SVEP and R ∈ B(X) is a Riesz operator which commutes with T , then T +R satisfies generalized a-Browder’s theorem. If, in particular, R is a quasi-nilpotent operator Q, then both T + Q and T ∗ + Q∗ satisfy generalized a-Browder’s theorem; furthermore, if Q is injective, then also T +Q satisfiesWeyl’s theorem. If A ∈ B(X) is an algebraic operator which commutes with the polynomially HP operator T , then T + N is polaroid and has SVEP, f (T + N) satisfies generalized Weyl’s theorem for every function f which is analytic on a neighbourhood of σ(T + N), and f (T + N)∗ satisfies generalized a-Weyl’s theorem for every function f which is analytic on, and constant on no component of, a neighbourhood of σ(T +N). © 2007 Elsevier Inc. All rights reserved