Classes of operators satisfying $a$-Weyl's theorem
In this article Weyl's theorem and $a$-Weyl's theorem on Banach spaces are related to an important property which has a leading role in local spectral theory: the single-valued extension theory. We show that if $T$ has SVEP then Weyl's theorem and $a$-Weyl's theorem for $T^\ast$ are equivalent, and analogously, if $T^\ast$ has SVEP then Weyl's theorem and $a$-Weyl's theorem for $T$ are equivalent. From this result we deduce that $a$-Weyl's theorem holds for classes of operators for which the quasi-nilpotent part $H_0(\lambda I-T)$ is equal to $\ker\, (\lambda I-T)^p$ for some $p\in \mathbb N$ and every $\lambda \in \mathbb C$, and for algebraically paranormal operators on Hilbert spaces. We also improve recent results established by Curto and Han, Han and Lee, and Oudghiri.