Perturbations of isometries between Banach spaces

Rafał Górak¹

¹ Technical University of Warsaw Pl. Politechniki 1 00-661 Warszawa, Poland

Studia Mathematica, Tome 207 (2011) no. 1, pp. 47-58

Résumé

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 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.

Détail

DOI : 10.4064/sm207-1-3

Keywords: prove general theorem concerning estimation expression frac frac different kinds maps satisfying general perturbed isometry condition seen quantitative generalization classical mazur ulam theorem estimates improve existing bi lipschitz maps consequence obtain simple proof result gevirtz which answers hyers ulam problem prove non linear generalization banach stone theorem which improves results jarosz recent results dutrieux kalton