Properties (t) and (gt) for Bounded Linear Operators

Abstract

In this paper we introduce and study the properties (t) and (gt), which extend properties (w) and (gw). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (t) and property (gt) hold. We also relate these properties with Weyl’s type theorems. We show that if T is a bounded linear operator acting on a Banach space \({\fancyscript{X}}\), then property (gt) holds for T if and only if property (gw) holds for T and σ(T) = σa(T). Analogously, we show that property (t) holds for T if and only if property (ω) holds for T and σ(T) = σa(T). We also study the properties (t) and (gt) for the operators satisfying the single valued extension property. Moreover, these properties are also studied in the framework of polaroid operators.