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 $\tx{\rm ind}(T-\lambdaI)=0$ for all $\lambdaın\sigma_a(T)\setminus\sigma_{SBF_+^-}(T)$;where $\sigma_{SBF_+^-}(T)$ 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 $E_a(T)=\Pi_a(T)$.