Hereditarily polaroid operators, SVEP and Weyl's theorem

Abstract

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 satisfies Weyl'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 ) .