Property (w) and perturbations

A bounded linear operator T∈L(X) defined on a Banach space X satisfies property (w), a variant of Weyl's theorem, if the complement in the approximate point spectrum σa(T) of the Weyl essential approximate spectrum σwa(T) coincides with 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 bounded operator T acting on a Banach space, under perturbations by finite rank operators, by nilpotent operator and quasi-nilpotent operators commuting with T.