Some results on isometric composition operators on Lipschitz 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 $C_{\varphi }:{\mathrm{Lip}}_0(M)⟶{\mathrm{Lip}}_0(N)$ is isometric depending on the properties of $\varphi$. We obtain a complete characterisation of those operators $C_{\varphi }$ in terms of a property of the function $\varphi$ 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 $C_{\varphi }$ being isometric in the case that $M$ is geodesic.

13 pages