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Abstract

Let $X$ be a Banach space and let $T$ be a bounded linear operator acting on $X$. Atkinson's well known theorem says that $T$ is a Fredholm operator if and only if its projection in the algebra $L(X)/ F_{0}(X)$ is invertible, where $ F_{0}(X)$ is the ideal of finite rank operators in the algebra $L(X)$ of bounded linear operators acting on $X$. In the main result of this paper we establish an Atkinson-type theorem for B-Fredholm operators. More precisely we prove that $T$ is a B-Fredholm operator if and only if its projection in the algebra $L(X)/ F_{0}(X)$ is Drazin invertible. We also show that the set of Drazin invertible elements in an algebra $A$ with a unit is a regularity in the sense defined by Kordula and Müller [8].