In this paper we introduce and study the new properties $(ab)$, $(gab)$, $(aw)$ and $(gaw)$ as a continuation of our previous article [4], where we introduced the two properties $(b)$ and $(gb)$. Among other, we prove that if $T$ is a bounded linear operator acting on a Banach space $X$, then $T$ possesses property $(gb)$ if and only if $T$ possesses property $(gab)$ and $\tx{\rm ind}(T-\lambda I)=0$ for all $ambdanigma_a(T)etminusigma_{SBF_+^-}(T)$; where $\sigma_{SBF_+^-}(T)$ is the essential semi-B-Fredholm spectrum of $T$ and $\sigma_a(T)$ is the approximate spectrum of $T$. We prove also that $T$ possesses property $(gaw)$ if and only if $T$ possesses property $(gab)$ and $E_a(T)=\Pi_a(T)$.