Continuous operators from spaces of Lipschitz functions

TL;DR

This paper investigates the existence and properties of continuous linear operators between Lipschitz function spaces and classical Banach spaces, revealing conditions under which such operators exist or fail, and characterizing the Schur property in Lipschitz-free spaces.

Contribution

It provides new criteria for the existence of continuous operators from Lipschitz spaces to classical Banach spaces and characterizes the Schur property for Lipschitz-free spaces.

Findings

No continuous surjections exist between certain Lipschitz and $C(K)$-spaces with weaker topologies.

If $M$ contains a bilipschitz copy of $S_{c_0}$, then $ ext{Lip}_0(M)$ admits a continuous operator onto $ ext{c}_0$ and $ ext{l}_1$.

A Lipschitz-free space has the Schur property iff it is weakly sequentially homeomorphic to all such spaces over discrete metric spaces of the same cardinality.

Abstract

We study the existence of continuous (linear) operators from the Banach spaces $\mbox{Lip}_0(M)$ of Lipschitz functions on infinite metric spaces $M$ vanishing at a distinguished point and from their predual spaces $\mathcal{F}(M)$ onto certain Banach spaces, including $C(K)$-spaces and the spaces $c_0$ and $\ell_1$. For pairs of spaces $\mbox{Lip}_0(M)$ and $C(K)$ we prove that if they are endowed with topologies weaker than the norm topology, then usually no continuous (linear or not) surjection exists between those spaces. It is also showed that if a metric space $M$ contains a bilipschitz copy of the unit sphere $S_{c_0}$ of the space $c_0$, then $\mbox{Lip}_0(M)$ admits a continuous operator onto $\ell_1$ and hence onto $c_0$. Using this, we provide several conditions for a space $M$ implying that $\mbox{Lip}_0(M)$ is not a Grothendieck space. Finally, we obtain a new characterization of the Schur property for Lipschitz-free spaces: a space $\mathcal{F}(M)$ has the Schur property if and only if for every complete discrete metric space $N$ with cardinality $d(M)$ the spaces $\mathcal{F}(M)$ and $\mathcal{F}(N)$ are weakly sequentially homeomorphic.