Variations on Weyl’s theorem

In this note we study the property (w), a variant ofWeyl’s theorem introduced by Rakoˇcevi´c, 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 (w) holds. We also relate this property with Weyl’s theorem and with another variant of it, a-Weyl’s theorem. We show that Weyl’s theorem, a-Weyl’s theorem and property (w) for T (respectively T ∗) coincide whenever T ∗ (respectively T ) satisfies SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property (w). © 2005 Elsevier Inc. All rights reserved.