Noncompactness and noncompleteness in isometries of Lipschitz spaces

TL;DR

This paper characterizes surjective linear isometries between Lipschitz function spaces, showing that completeness of the underlying metric spaces is not necessary for isometries to be weighted composition maps, especially when the codomain spaces are not complete.

Contribution

It provides a comprehensive characterization of isometries in Lipschitz spaces, including cases where the underlying metric spaces are not complete, extending previous scalar results.

Findings

All isometries are weighted composition maps under certain conditions.

Completeness of metric spaces is not required for isometry characterization in non-scalar cases.

The general form of isometries is established, both weighted and non-weighted.

Abstract

We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions $\mathrm{Lip}(X,E)$ and $\mathrm{Lip}(Y,F)$, for strictly convex normed spaces $E$ and $F$ and metric spaces $X$ and $Y$: \begin{enumerate} \item Characterize those base spaces $X$ and $Y$ for which all isometries are weighted composition maps. \item Give a condition independent of base spaces under which all isometries are weighted composition maps. \item Provide the general form of an isometry, both when it is a weighted composition map and when it is not. \end{enumerate} In particular, we prove that requirements of completeness on $X$ and $Y$ are not necessary when $E$ and $F$ are not complete, which is in sharp contrast with results known in the scalar context.