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

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