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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 Lipschitzmap. This improves 25 years old Casazza-Kalton-Christensen-van Eijndhovenperturbation. It also improves 28 years old Soderlind-Campanato perturbationand 2 years old Barbagallo-Ernst-Thera perturbation. We give applications tothe theory of metric frames. The notion of Lipschitz atomic decomposition forBanach spaces is also introduced.

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