On non-adjointable semi-C*-Fredholm operators and semi-C*-Weyl operators

Abstract

We extend the results from semi-Fredholm theory of adjointable, bounded C*-operators on the standard C*-module, presented in [3], to the case of general bounded C*-operators on arbitrary Hilbert C*-modules. Next, in the special case of the standard C*-module, we show that the set of those semi-C*-Fredholm operators that are not semi-C*-Weyl operators is open in the norm topology, and that the set of non-adjointable semi-C*-Weyl operators is invariant under perturbations by general compact operators. Moreover, we provide an extended Schechter characterization and a generalized Fredholm alternative in the case of adjointable C*-operators on the standard C*-module. Finally, we provide examples of semi-C*-Fredholm operators.