Property (w) and perturbations III

The property (w) is a variant of Weylʹs theorem, for a bounded operator T acting on a Banach space. In this note we consider the preservation of property (w) under a finite rank perturbation commuting with T, whenever T is polaroid, or T has analytical core K ( λ 0 I − T ) = { 0 } for some λ 0 ∈ C . The preservation of property (w) is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations. The theory is exemplified in the case of some special classes of operators.