Index of B-Fredholm operators and generalization of a Weyl theorem

The aim of this paper is to show that if $S$ and $T$ are commuting B-Fredholm operators acting on a Banach space $X$, then $ST$ is a B-Fredholm operator and $ind(ST)=ind(S)+ind(T)$, where $ind$ means the index. Moreover if $T$ is a B-Fredholm operator and $F$ is a finite rank operator, then $T+F$ is a B-Fredholm operator and $ind(T+F)= ind(T).$ We also show that if $0$ is isolated in the spectrum of $T$, then $T$ is a B-Fredholm operator of index $0$ if and only if $T$ is Drazin invertible. In the case of a normal bounded linear operator $T$ acting on a Hilbert space $H$, we obtain a generalization of a classical Weyl theorem.