Compact and weakly compact Lipschitz operators
Résumé
Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $f : \mathcal F (M) \to \mathcal F (N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for f to be compact in terms of metric conditions on $f$. This extends a result by A. Jiménez-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 f is compact if and only if it is weakly compact.
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