Properties $(S)$ and $(gS)$ for Bounded Linear Operators

Abstract

An operator $T$ acting on a Banach space $\X$ obeys property $(R)$ if $\pi_a^0(T)=E^0(T),$ where $\pi_a^0(T)$ is the set of all left poles of $T$ of finite rank and $E^0(T)$ is the set of all isolated eigenvalues of $T$ of finite multiplicity. In this paper we introduce and study two new properties $(S)$ and $(gS)$  in connection with Weyl type theorems. Among other things, we prove that if $T$ is a bounded linear operator acting on a Banach space, then $T$ satisfies property $(R)$ if and only if $T$ satisfies property $(S)$ and $\pi^0(T)=\pi_a^0(T),$ where $\pi^0(T)$ is the set of poles of finite rank. Also we show that   Weyl's theorem holds for $T$ if and only if  $T$ satisfies property $(S)$ and $\sigma_w(T)=\sigma_b(T)$, where $\sigma_w(T)$ is the Weyl spectrum and $\sigma_b(T)$ is the the Browder spectrum. Analogous results for property $(gS)$ are given. Moreover, these properties are also studied in the frame of polaroid operator.