Chain-finite operators and locally chain-finite operators.
The problem we are concerned with in this research announcement is the algebraic characterization of chain-finite operators (global case) and of locally chain-finite operators (local case).
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 H₀(λI...