Property (gz) for bounded linear operators

Abstract

A bounded linear operator $T$ acting on a Banachspace possesses property (gaw) if $\sigma(T)\setminusE_a(T)=\sigma_{BW}(T)$, where $\sigma_{BW}(T)$ is the B-Weylspectrum of $T$, $\sigma(T)$ is the usual spectrum of $T$ and$E_a(T)$ is the set of all eigenvalues of $T$ which are isolatedin the approximate point spectrum of $T$. In this paper weintroduce and study the new spectral properties (z), (gz), (az)and (gaz) as a continuation of [M. Berkani, H. Zariouh, {ıt New extended Weyl type theorems},Mat. Vesnik {\bf 62} (2010), 145–154], which are related toWeyl type theorems. Among other results, we prove that $T$possesses property (gz) if and only if $T$ possesses property(gaw) and $\sigma_{BW}(T)=\sigma_{SBF_+^-}(T)$; where$\sigma_{SBF_+^-}(T)$ is the essential semi-B-Fredholm spectrum of $T$.