New extended Weyl type theorems

Abstract

In this paper we introduce and study the new properties $(ab)$,$(gab)$, $(aw)$ and $(gaw)$ as a continuation of our previousarticle [4], where we introduced the two properties $(b)$ and$(gb)$.Among other, we prove that if $T$ is a bounded linear operatoracting on a Banach space $X$, then $T$ possesses property $(gb)$ ifand only if $T$ possesses property $(gab)$ and $\text{tx}\mathrm{i}\mathrm{n}\mathrm{d}(T-\text{lambdaI})=0$ for all $\lambda ın{\sigma }_a(T)\setminus {\sigma }_{SB{F}_{+}^{-}}(T)$;where ${\sigma }_{SB{F}_{+}^{-}}(T)$ is the essential semi-B-Fredholmspectrum of $T$ and ${\sigma }_a(T)$ is the approximate spectrum of$T$. We prove also that $T$ possesses property $(gaw)$ if and onlyif $T$ possesses property $(gab)$ and $E_a(T)={\mathrm{\Pi }}_a(T)$.