WEYL'S THEOREM, $a$-WEYL'S THEOREM, AND LOCAL SPECTRAL THEORY

Abstract

Necessary and sufficient conditions are given for a Banach space operator with the single-valued extension property to satisfy Weyl's theorem and a-Weyl's theorem. It is shown that if T or T* has the single-valued extension property and T is transaloid, then Weyl's theorem holds for f(T)for every f∈H(σ(T)). When T* has the single-valued extension property, T is transaloid and T is a-isoloid, then a-Weyl's theorem holds for f(T) for every f∈H(σ(T)). It is also proved that if T or T* has the single-valued extension property, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.