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

Abstract

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 Schechter—Lebow 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.