Compact and weakly compact Lipschitz operators

TL;DR

This paper characterizes when Lipschitz operators between metric spaces extend to compact operators on Lipschitz-free spaces, linking metric conditions with operator compactness and weak compactness.

Contribution

It provides a necessary and sufficient metric condition for the compactness of Lipschitz operator extensions, extending previous results to non-separable and unbounded spaces.

Findings

Characterization of compactness via metric conditions

Equivalence of compactness and weak compactness for these operators

Extension of results to non-separable and unbounded metric spaces

Abstract

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat{f} : \mathcal F(M) \to \mathcal F(N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for $\widehat{f}$ to be compact in terms of metric conditions on $f$. This extends a result by A. Jim\'{e}nez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behavior of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that $\widehat{f}$ is compact if and only if it is weakly compact.