On Operators with Closed Range and Semi-Fredholm Operators Over W*-Algebras

In this paper, we consider A- Fredholm and semi-A- Fredholm operators on Hilbert C *-modules over a W *-algebra A defined in [3] and [9]. Using the assumption that A is a W *-algebra (rather than an arbitrary C *-algebra), we obtain a generalization of SchechterLebow characterization of semi-Fredholm operators and a generalization of the "punctured neighborhood" theorem, as well as some other results generalizing their classical counterparts. We consider both adjointable and nonadjointable semi-Fredholm operators over W *-algebras. Moreover, we also work with general bounded adjointable operators with closed ranges over C *-algebras and prove a generalization of a Bouldin result for Hilbert spaces to Hilbert C *-modules.