Generalized stability of isometries
type: Item
instance of: scholarly article
title: Generalized stability of isometries
zbMATH Open document ID: 0956.46007
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-03-02T00:00:00Z
review text: Let \(X\), \(Y\) be two Banach spaces. The author introduces the following concept of isometry: the map \(f:X\rightarrow Y\) is an \((e,p)\)-isometry if \[ ||f(x)-f(y)|-|x-y||\leq e|x-y|^p \] holds for all \(x,y \in X\), and studies the \(p\)-stability of a couple of Banach spaces \(X,Y\) with respect to this concept. (The pair of Banach space \((X,Y)\) is \(p\)-stable if for every surjective \((e,p)\)-isometry \(f:X\rightarrow Y\) there exists a surjective isometry \(U:X\rightarrow Y\) satisfiying the inequality \(|f(x)-U(x)|\leq \delta(e)|x|^p\) for all \(x\in X\) and a certain fixed positive function \(\delta(e)\).) It is proved that every pair of Banach spaces \((X,Y)\) is \(p\)-stable for all \(p\in [0,1)\) although \((\mathbb{R}^2,\mathbb{R}^2)\) is a counterexample for \(1-\)stability. A superstability result is also proved for pair of finite dimensional Banach spaces and \(p>1\). The paper is very nice, and is connected to previous work of other people. Another interesting and easy to read paper on the subject is the following: \textit{R. Bhatia} and \textit{P. Šemrl} [Am. Math. Mon. 104, No.~6, 497-504 (1997; Zbl 0901.46016)].
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 1423637
zbMATH Keywords: \((e,p)\)-isometry
zbMATH Keywords: \(p\)-stable
zbMATH Keywords: stability of isometries
zbMATH Keywords: superstability
author: Gregor Dolinar
reviewed by: Jose Maria Almira
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6649
OpenAlex ID: W2000483063
cites work: Approximate Isometries on Euclidean Spaces
cites work: Approximate isometries
cites work: Uniformly Convex Spaces
cites work: Stability of Isometries on Banach Spaces
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cites work: On approximate isometries
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cites work: On quasi‐isometric mappings, I
cites work: Q3711063
cites work: Q4333558
cites work: On nonlinear perturbations of isometries
cites work: Approximately linear functionals on Hilbert spaces
cites work: Hyers-Ulam stability for isometries on Banach spaces
DOI: 10.1006/JMAA.1999.6649
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description: scientific article; zbMATH DE number 1423637