Weyl's theorem, a-Weyl's theorem, and local spectral theory

Raul E. Curto, Young Min Han

Published 2002-07-06Version 1

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 $T^{\ast}$ has SVEP and $T$ is transaloid, then Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma (T))$. When $T^{\ast}$ has SVEP, $T$ is transaloid and $T$ is $a$-isoloid, then $a$-Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma (T))$. We also prove that if $T$ or $T^{\ast}$ has SVEP, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.