In this paper we intro duce 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 ind(T . λI) = 0 for all λ ∈ σ a (T ) σ SBF . + (T ); where σ SB F . + (T ) is the essential semi-B-Fredholm spectrum of T and σ 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 ) = Π a (T ).
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