Compact and weakly compact Lipschitz operators

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. 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.

On the Dynamics of Lipschitz Operators

“…Since ε was arbitrary, we obtain λ = 0. □ Since when f : F(M ) → F(M ) is compact, f is automatically radially flat (see Lemma 2.4 in [2]), we obtain the following corollary. Corollary 3.10.…”

“…Next, we wish to highlight that w f and wC f are in fact compact if and only if they are weakly compact. This was already known in the real case when w ≡ 1, see [2,Theorem B]. We use the same arguments to derive the case of weighted operators.…”

“…Hence, in this paper we shall study w f and deduce the desired properties of wC f as simple consequences. We obtain in this way some refinements of the results mentioned above by dropping as many assumptions as possible on M and w. At this point, we wish to point out that this strategy was followed in [2] for the simpler case w = 1. It allowed the authors to remove the assumptions of separability and boundedness of M in the characterisation of compactness obtained in [21].…”

“…A full characterization of compact Lipschitz operators (w ≡ 1) is provided in [2]; it is the natural continuation of the works in [9,15]. For compactness results in the weighted case, we refer the reader to [3,8] (and references therein).…”

“…Another important ingredient is Theorem 2.14 which allows us to prove that weak compactness is actually equivalent to (norm) compactness for the class of operators we consider in this article. The latter result in an improvement of [2,Theorem B] where w = 1, and [19,Theorem 2.1] where M is a compact purely 1-unrectifiable metric space. The two key elements mentioned above permit us to deduce a new and general characterization of (weak) compactness for weighted Lipschitz/composition operators.…”