# On operators with closed range and semi-Fredholm operators over W*-algebras

## TL;DR

This paper extends classical results on semi-Fredholm operators to the setting of Hilbert C*-modules over W*-algebras, generalizing key theorems and characterizations in operator theory.

## Contribution

It provides new generalizations of semi-Fredholm operator characterizations and the punctured neighborhood theorem specifically for W*-algebra modules, including both adjointable and non-adjointable cases.

## Key findings

- Generalized Schechter-Lebow characterization for semi-Fredholm operators
- Extended punctured neighborhood theorem to W*-algebra context
- Proved results for bounded, adjointable operators with closed range over C*-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],[10]. Using the assumption that A is a W*-algebra (and not an arbitrary C*-algebra), we obtain several results such as generalization of Schechter-Lebow characterization of semi-Fredholm operators and generalization of "punctured neighbourhood" theorem, as well as some other results that generalize their classical counterparts. We consider both adjointable and non adjointable semi-Fredholm operators over W*-algebras. Moreover, we also work with general bounded, adjointable operators with closed range over C*-algebras and prove a generalization to Hilbert C*-modules of the result in [1] on Hilbert spaces.