Isometric Composition Operators on Lipschitz Spaces

Abstract

Given pointed metric spaces X and Y, we characterize the basepoint-preserving Lipschitz maps $$\phi $$ from Y to X inducing an isometric composition operator $$C_\phi $$ between the Lipschitz spaces $$\mathrm {Lip}_0(X)$$ and $$\mathrm {Lip}_0(Y)$$, whenever X enjoys the peak property. This gives an answer to a question posed by Weaver in his book [Lipschitz algebras. Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018].