Perturbations of isometries between Banach spaces

Rafał Górak. "Perturbations of isometries between Banach spaces." Studia Mathematica 207.1 (2011): 47-58. <http://eudml.org/doc/285589>.

@article{RafałGórak2011,
abstract = {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.},
author = {Rafał Górak},
journal = {Studia Mathematica},
keywords = {Mazur-Ulam theorem; Banach-Stone theorem; function space; isometry},
language = {eng},
number = {1},
pages = {47-58},
title = {Perturbations of isometries between Banach spaces},
url = {http://eudml.org/doc/285589},
volume = {207},
year = {2011},
}

TY - JOUR
AU - Rafał Górak
TI - Perturbations of isometries between Banach spaces
JO - Studia Mathematica
PY - 2011
VL - 207
IS - 1
SP - 47
EP - 58
AB - 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.
LA - eng
KW - Mazur-Ulam theorem; Banach-Stone theorem; function space; isometry
UR - http://eudml.org/doc/285589
ER -