Improving Casazza-Kalton-Christensen-van Eijndhoven Perturbation with Applications

TL;DR

This paper improves longstanding perturbation results for invertible Lipschitz maps between Banach spaces, providing new conditions ensuring invertibility and applications to metric frames and Lipschitz atomic decompositions.

Contribution

It advances 25-year-old perturbation theory for invertible Lipschitz maps, extending previous results and introducing applications to metric frames and Lipschitz atomic decompositions.

Findings

Proves invertibility of certain Lipschitz maps under new conditions.

Enhances classical perturbation theorems by Casazza-Kalton-Christensen-van Eijndhoven and others.

Introduces the concept of Lipschitz atomic decomposition for Banach spaces.

Abstract

Let $ \mathcal{X}$, $ \mathcal{Y}$ be Banach spaces and $S:\mathcal{X} \to \mathcal{Y} $ be an invertible Lipschitz map. Let $ T : \mathcal{X}\rightarrow \mathcal{Y}$ be a map and there exist $ \lambda_1,\lambda_2 \in \left [0, 1 \right )$ such that \begin{align*} \|Tx-Ty-(Sx-Sy)\|\leq\lambda_1\|Sx-Sy\|+\lambda_2\|Tx-Ty\|,\quad \forall x,y \in \mathcal{X}. \end{align*} Then we prove that $T$ is an invertible Lipschitz map. This improves 25 years old Casazza-Kalton-Christensen-van Eijndhoven perturbation. It also improves 28 years old Soderlind-Campanato perturbation and 2 years old Barbagallo-Ernst-Thera perturbation. We give applications to the theory of metric frames. The notion of Lipschitz atomic decomposition for Banach spaces is also introduced.