On Drazin invertible C* -operators and generalized C*-Weyl operators

Abstract

Generalized Weyl operators on Hilbert spaces have been introduced and studied by Djordjević (Proc Am Math Soc, 130:81–ll4, 2001). In this paper, we provide a generalization of his result in the setting of C*-operators on Hilbert C*-modules by giving sufficient conditions under which the sum of a generalized C*-Weyl operator and a finitely generated C*-operator is a generalized C*-Weyl operator. Also, we obtain an extension of Djordjević’s results from the case of operators on Hilbert spaces to the case of operators on Banach spaces. Next, we consider semi-C*-B-Fredholm operators on Hilbert C*-modules and give sufficient conditions under which the composition of two mutually commuting semi-C*-B-Fredholm operators is a semi-C*-B-Fredholm operator, thus generalizing the result by Berkani regarding semi-B-Fredholm operators on Banach spaces. Finally, we consider Drazin invertible C*-operators, and we give necessary and sufficient conditions for two mutually commuting C*-operators to be Drazin invertible when their composition is Drazin invertible.