  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.
