**Mohammed Berkani**

*University Mohammed I*

Title: B-Fredholm theory and Drazin invertibility

**Abstract:** In this talk we will introduce the class of B-Fredholm operators, as a natural generalization of Fredholm operators. While the invertibility in the Calkin algebra characterizes Fredholm operators, we show that Drazin invertibility in the algebra L(X)/ F_0(X), F_0(X) being the ideal of finite rank operators, characterizes B-Fredholm operators. Then we will answer a natural arising question: What are the properties of those operators whose image under the canonical homomorphism from L(X) onto L(X)/ F_0(X) is generalized Drazin invertible?