Some results on isometric composition operators on Lipschitz spaces

TL;DR

This paper characterizes when isometric composition operators between Lipschitz spaces are isometric, based on properties of the underlying functions and metric spaces, especially in geodesic cases.

Contribution

It provides a complete characterization of isometric composition operators on Lipschitz spaces under specific geometric conditions of the metric spaces.

Findings

Characterization of isometric composition operators when $B_{\\mathcal F(M)}$ is the convex hull of preserved extreme points.

Necessary and sufficient conditions for isometry when $M$ is geodesic.

Results depend on properties of the function $\phi$ and the structure of the metric spaces.

Abstract

Given two metric spaces $M$ and $N$ we study, motivated by a question of N. Weaver, conditions under which an isometric composition operator $C_\phi:\mathrm{Lip}_0(M)\longrightarrow \mathrm{Lip}_0(N)$ is isometric depending on the properties of $\phi$. We obtain a complete characterisation of those operators $C_\phi$ in terms of a property of the function $\phi$ in the case that $B_{\mathcal F(M)}$ is the closed convex hull of its preserved extreme points. Also, we obtain necessary and sufficient conditions for $C_\phi$ being isometric in the case that $M$ is geodesic.