A note on isometries of Lipschitz spaces
type: Item
instance of: scholarly article
title: A note on isometries of Lipschitz spaces
zbMATH Open document ID: 1341.46019
author: Alireza Ranjbar-Motlagh
published in: Journal of Mathematical Analysis and Applications
publication date: 2016-01-07T00:00:00Z
review text: This paper deals with spaces of vector-valued Lipschitz functions. Let \((V, \|\cdot \|)\) and \((W, \|\cdot \|)\) be real or complex Banach spaces and let \(X\) and \(Y\) be compact metric spaces. The spaces \(\mathrm{Lip}(X, V)\) and \(\mathrm{Lip}(Y,W)\) consist of all Lipschitz functions defined on \(X\) and \(Y\), with values in \(V\) and \(W\), respectively. These spaces are endowed with the norm \(\|f\|= \max \{ \mathcal{L}(f), \|f\|_{\infty}\},\) where \(\mathcal{L}(f)\) is the Lipschitz constant of \(f\). The map \(T: \mathrm{Lip}(X, V) \rightarrow \mathrm{Lip}(Y, W)\) has property \(Q\) if, for every \(y \in Y\) and \(w \in W\), there exists \(v \in V\) such that \(T \hat{v} (y)=w\), where \(\hat{v}\) denotes the constant map in \(\mathrm{Lip}(X, V)\) equal \(v\). The statement of the main theorem in this paper generalizes a result in [\textit{F. Botelho} et al., J. Math. Anal. Appl. 381, No. 2, 821--832 (2011; Zbl 1228.46020)] by removing a quasi-sub-reflexivity condition on the range spaces. { Theorem.} Let \(X\) and \(Y\) be compact metric spaces and let \(V\) and \(W\) be Banach spaces with trivial centralizers. Let \(T: \mathrm{Lip}(X, V) \rightarrow \mathrm{Lip}(Y, W)\) be a surjective linear isometry such that both \(T\) and \(T^{-1}\) have property \(Q\), then \(T\) is a weighted composition operator of the form \[ Tf(y) =J (y) f(\phi(y)) \] for all \(f \in \mathrm{Lip} (X, V)\) and \(y \in Y\), where \(\phi : Y\rightarrow X\) is a bi-Lipschitz homeomorphism and \(J\) is a Lipschitz map from \(Y\) into the space of surjective linear isometries from \(V\) into \(W\). The author raises the question of whether it is possible to relax the compactness assumption on the metric spaces.
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 26A16
zbMATH DE Number: 6527119
zbMATH Keywords: isometry
zbMATH Keywords: vector-valued Lipschitz function
zbMATH Keywords: Banach-Stone theorem
reviewed by: Fernanda Botelho
MaRDI profile type: Publication
full work available at URL: j.jmaa.2013.10.003
OpenAlex ID: W2022234958
DOI: 10.1016/J.JMAA.2013.10.003
description: scientific article; zbMATH DE number 6527119