WEYL’S THEOREM, a-WEYL’S THEOREM, AND LOCAL SPECTRAL THEORY

We give necessary and sufficient conditions for a Banach space operator with the single valued extension property (SVEP) to satisfy Weyl’s theorem and a-Weyl’s theorem. We show that if T or T ∗ has SVEP and T is transaloid, then Weyl’s theorem holds for f(T ) for every f ∈ H(σ(T )). When T ∗ has SVEP, T is transaloid and T is a-isoloid, then a-Weyl’s theorem holds for f(T ) for every f ∈ H(σ(T )). We also prove that if T or T ∗ has SVEP, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.