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 [IS1], 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 non-adjointable 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.