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 $T^{\ast }$ has SVEP and $T$ is transaloid, then Weyl’s theorem holds for $f(T)$ for every $f\in H(\sigma (T))$\mathrm{.}When$T^{\ast}$hasSVEP,$T$istransaloidand$T$is$a$-isoloid,then$a$-Wey{l}^{\mathrm{\prime }}stheoremholdsfor$f(T)$forevery$f\in H(\sigma (T))$. 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.