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. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behaviour 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.
Any Lipschitz map f : M → N between two pointed metric spaces may be extended in a unique way to a bounded linear operator f : F (M ) → 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.
We consider weighted composition operators, that is operators of the type g → w • g • f , acting on spaces of Lipschitz functions. Bounded weighted composition operators, as well as some compact weighted composition operators, have been characterized quite recently. In this paper, we provide a different approach involving their pre-adjoint operators, namely the weighted Lipschitz operators acting on Lipschitz free spaces. This angle allows us to improve some results from the literature. Notably, we obtain a distinct characterization of boundedness with a precise estimate of the norm. We also characterise injectivity, surjectivity, compactness and weak compactness in full generality.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.