Polaroid Operators with SVEP and Perturbations of Property (gw)

Let ${\mathcal{L}(X)}$ be the algebra of all bounded linear operators on X and ${\mathcal{P}S(X)}$ be the class of polaroid operators with the single-valued extension property. The property (gw) holds for ${T \in \mathcal{L}(X)}$ if the complement in the approximate point spectrum of the semi-B-essential approximate point spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues of the spectrum. In this note we focus on the stability of the property (gw) under perturbations: we prove that, if ${T \in \mathcal{P}S(X)}$ and A (resp. Q) is an algebraic (resp. quasinilpotent) operator, then the property (gw) holds for f(T ^* + A ^*) (resp. f(T ^* + Q*)) for every analytic function f in σ(T + A) (resp. σ(T + Q)). Some applications are also given.