Weakly compact composition operators on spaces of Lipschitz functions

TL;DR

This paper proves that under certain conditions, weakly compact composition operators on Lipschitz function spaces are actually compact, clarifying the relationship between these operator types.

Contribution

It establishes that weakly compact composition operators on Lipschitz function spaces are necessarily compact when the space has the uniform separation property.

Findings

Weakly compact composition operators are compact under specified conditions.

The result applies to spaces of Lipschitz functions on pointed compact metric spaces.

The uniform separation property is key to the main theorem.

Abstract

Let $X$ be a pointed compact metric space. Assuming that $\mathrm{lip}_0(X)$ has the uniform separation property, we prove that every weakly compact composition operator on spaces of Lipschitz functions $\mathrm{Lip}_0(X)$ and $\mathrm{lip}_0(X)$ is compact.