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arXiv · 2109.02127

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

K. Mahesh Krishna
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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.

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K. Mahesh Krishna. 2021-09-05. Improving Casazza-Kalton-Christensen-van Eijndhoven Perturbation with Applications. https://doi.org/10.7153/jmi-2024-18-10

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