# Perturbations of isometries between Banach spaces

Studia Mathematica (2011)

- Volume: 207, Issue: 1, page 47-58
- ISSN: 0039-3223

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topRafał 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 -

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