Nonvanishing Preservers and Compact Weighted Composition Operators between Spaces of Lipschitz Functions

Abstract

We will give the $\alpha$-Lipschitz version of the Banach-Stone type theorems for lattice-valued $\alpha$-Lipschitz functions on some metric spaces. In particular, when $X$ and $Y$ are bounded metric spaces, if $T:\text{L}\text{i}\text{p}\left(X\right)\to \text{L}\text{i}\text{p}\left(Y\right)$ is a nonvanishing preserver, then $T$ is a weighted composition operator $Tf=h·f\circ \phi$, where $\phi :Y\to X$ is a Lipschitz homeomorphism. We also characterize the compact weighted composition operators between spaces of Lipschitz functions.