Property (aw) and Weyl’s Theorem

Abstract

In this note we study the property (aw), a variant of Weyl’s theorem introduced by Berkani and Zariouh, by means of the localized single valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (aw) holds. We also relate this property with Weyl’s theorem, a-Weyl’s theorem and property (w). Finally, we show that if T is a-polaroid and either T or T* has SVEP then f(T) satisfies property (aw) for each \({f \in H_1(\sigma(T))}\) .