WEYLS THEOREM, alpha-WEYLS THEOREM, AND LOCAL SPECTRAL THEORY

Necessary and sufficient conditions are given for a Banach space operator with the single-valued extension property to satisfy Weylʹs theorem and alpha-Weylʹs theorem. It is shown that if T or T* has the single-valued extension property and T is transaloid, then Weylʹs theorem holds for f(T) for every f epsilonin H(\sigma (T)). When T* has the single-valued extension property, T is transaloid and T is alpha-isoloid, then alpha-Weylʹtheorem holds for f(T) for every f epsilon H(sigma(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.