We prove a very general theorem concerning the estimation of the expression ||T((a+b)/2) - (Ta+Tb)/2|| for different kinds of maps T satisfying some general perturbed 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm207-1-3,
author = {Rafa\l\ G\'orak},
title = {Perturbations of isometries between Banach spaces},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {47-58},
zbl = {1250.46007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm207-1-3}
}
Rafał Górak. Perturbations of isometries between Banach spaces. Studia Mathematica, Tome 204 (2011) pp. 47-58. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm207-1-3/