Property $$(UW_\Pi )$$ and Perturbations

Abstract

Property \((UW_\Pi )\) holds for a bounded linear operator \(T \in B(X),\) defined on a complex infinite dimensional Banach space X, if the poles of the resolvent of T are exactly the spectral points \(\lambda \) for which \(\lambda I-T\) is upper semi-Weyl. In this paper, we discuss the relationship between property \((UW_\Pi )\) and other Weyl type theorems. The stability of property \((UW_\Pi )\) is also studied under nilpotent, quasi-nilpotent, finite-dimensional or compact perturbations commuting with T.