Property (w) for perturbations of polaroid operators

A bounded linear operator T∈L(X) acting on a Banach space satisfies property (w), a variant of Weyl’s theorem, if the complement in the approximate point spectrum σa(T) of the Weyl essential approximate-point spectrum σwa(T) is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this note, we study the stability of property (w) for a polaroid operator T acting on a Banach space, under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic operators commuting with T.