# Stability of Fredholm property for regular operators on Hilbert $C^*$-modules

## TL;DR

This paper investigates the stability of the Fredholm property for regular operators on Hilbert $C^*$-modules under various perturbations, establishing openness in the gap metric and constructing continuous paths of such operators.

## Contribution

It proves the stability of the Fredholm property under bounded, compact, and gap metric perturbations, and shows the space of regular Fredholm operators is open in the operator space.

## Key findings

- Fredholm property is stable under certain perturbations.
- The space of regular Fredholm operators is open in the gap metric topology.
- Constructs continuous paths of selfadjoint regular Fredholm operators.

## Abstract

We study the stability of Fredholm property for regular operators on Hilbert $C^*$-modules under some certain perturbations. We treat this problem when perturbing operators are (relatively) bounded or relatively compact. We also consider the perturbations of regular Fredholm operators in terms of the gap metric. In particular, we prove that the space of all regular Fredholm operators on a Hilbert $C^*$-module $E$ is open in the space of all regular operators on $E$ with respect to the gap metric. As an application, we construct some continuous paths of selfadjoint regular Fredholm operators with respect to the gap metric.