# Semi-Fredholm Theory on Hilbert C*-modules

## TL;DR

This paper extends the classical semi-Fredholm theory to Hilbert C*-modules, defining semi-Fredholm operators and proving their properties, including one-sided invertibility modulo compact operators and openness of the set.

## Contribution

It introduces a new semi-Fredholm theory on Hilbert C*-modules, generalizing existing Fredholm theory and establishing foundational properties of semi-Fredholm operators in this context.

## Key findings

- Semi-Fredholm operators are characterized as one-sided invertible modulo compact operators.
- The set of proper semi-Fredholm operators is shown to be open.
- The theory generalizes classical results to the setting of Hilbert C*-modules.

## Abstract

In this paper we establish the semi-Fredholm theory on Hilbert C*-modules as a continuation of Fredholm theory on Hilbert C*-modules established by Mishchenko and Fomenko. We give a definition of a semi-Fredholm operator on Hilbert C*-module and prove that these semi-Fredholm operators are those that are one-sided invertible modulo compact operators, that the set of proper semi-Fredholm operators is open and many other results that generalize their classical counterparts.