Perturbations of isometries between Banach spaces

TL;DR

This paper provides a general theorem estimating deviations of maps from isometries in Banach spaces, extending classical results and improving bounds for bi-Lipschitz maps, with applications to the Hyers-Ulam problem and Banach-Stone theorem generalizations.

Contribution

It introduces a broad theorem that generalizes the Mazur-Ulam theorem and improves existing estimates for perturbed isometries, with significant applications.

Findings

Improved estimates for bi-Lipschitz maps.

Simplified proof of Gevirtz's result on Hyers-Ulam problem.

Non-linear generalization of the Banach-Stone theorem.

Abstract

We prove a very general theorem concerning the estimation of the expression $\|T(\frac{a+b}{2}) - \frac{Ta+Tb}{2}\|$ for different kinds of maps $T$ satisfying some general perurbated isometry condition. It can be seen as a quantitative generalization of the classical Mazur-Ulam theorem. The estimates improve the existing ones for bi-Lipschitz maps. As a consequence we also obtain a very simple proof of the result of Gevirtz which answers the Hyers-Ulam problem and we prove a non-linear generalization of the Banach-Stone theorem which improves the results of Jarosz and more recent results of Dutrieux and Kalton.