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arXiv · 2110.03231

Compact and weakly compact Lipschitz operators

Arafat Abbar·Clément Coine·Colin Petitjean
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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.

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Arafat Abbar, Clément Coine, Colin Petitjean. 2021-10-07. Compact and weakly compact Lipschitz operators. https://arxiv.org/abs/2110.03231

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