A note on isometries of Lipschitz spaces
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.
scientific article; zbMATH DE number 6527119