Perturbations of isometries between Banach spaces

Abstract

We prove a very general theorem concerning the estimation of the expression $\Vert T(\frac{a+b}{2})-\frac{Ta+Tb}{2}\Vert$ 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.