Weyl's Theorem, a-Weyl's Theorem, and Local Spectral Theory

Necessary and sufficient conditions are given for a Banach spaceoperator with the single-valued extension property to satisfyWeyl's theorem and a-Weyl's theorem. It is shown that if T orT^* has the single-valued extension property and T is transaloid,then Weyl's theorem holds for f(T)for every fH((T)). When T^*has the single-valued extension property, T is transaloid andT is a-isoloid, then a-Weyl's theorem holds for f(T) for everyfH((T)). It is also proved that if T or T^* has the single-valuedextension property, then the spectral mapping theorem holdsfor the Weyl spectrum and for the essential approximate pointspectrum.