Property (t) and perturbations

Abstract

In this paper we study the properties of property (t), which is introduced by Rashid. We investigate the property (t) in connection with Weyl type theorems, and establish sufficient and necessary conditions for which property (t) holds. Especially, we obtain the equivalence of a-Weyl’s theorem and property (t) without the condition that T is a-polaroid, which improves a corresponding result of Rashid (Mediterr J Math 11:1–16, 2014). We also study the stability of property (t) under perturbations by nilpotent operators, by finite rank operators, by quasi-nilpotent operators and by Riesz operators commuting with T.