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* are equivalent, and analogously, if T* 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(λI-T) is equal to ker (λI -T)p for some p ∈ ℕ and every λ ∈ ℂ, and for algebraically paranormal operators on Hilbert spaces. We also improve recent results established by Curto and Han, Han and Lee, and Oudghiri.