This is an announcement for the paper "Perturbations of isometries between Banach spaces" by Rafal Gorak.
Abstract: We prove a very general theorem concerning the estimation of the expression \mbox{$|T(\frac{a+b}{2}) - \frac{Ta+Tb}{2}|$} 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.
Archive classification: math.FA
Mathematics Subject Classification: 46E40, 46B20
Submitted from: R.Gorak@mini.pw.edu.pl
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1105.0854
or