Isometric Composition Operators on Lipschitz Spaces

Abstract

Given pointed metric spaces X and Y , we characterize the basepoint-preserving Lipschitz maps φ from Y to X inducing an isometric composition operator C φ between the Lipschitz spaces Lip 0 (X) and Lip 0 (Y ), whenever X enjoys the peak property. This gives an answer to a question posed by N. Weaver in his book [Lipschitz algebras. Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018].

Some results on isometric composition operators on Lipschitz spaces

“…The key ingredient for obtaining this result is that, in Lipschitz-free Banach spaces, a molecule m x,y is preserved extreme if, and only if, it is denting (see the subsection Notation for details and references). Thus the equivalence established in [14,Theorem 2.4] actually works for a rather larger class of metric spaces M (see Example 2.4). While in Sect.…”

“…→ 1, and we are done. Thus, we can obtain the announced generalisation of [14,Theorem 2.4]. (3) If M is boundedly compact (i.e.…”

“…Notice that, trivially, such φ must be of norm one and with dense range. Very recently, A. Jiménez-Vargas obtained in [14] a characterisation in the following sense: if φ : N −→ M is a norm-one Lipschitz function and M has the so-called peaking property, then C φ : Lip 0 (M) −→ Lip 0 (N ) is an isometry if, and only if, for every pair of points x, y ∈ M, x = y we can find sequences x n ⊆ N and y n ⊆ N such that φ(x n ) → x, φ(y n ) → y and…”

“…With the previous information in mind, the main aim of Section 2 is to generalise the above mentioned result [14,Theorem 2.4] and to prove that, if B F (M) is the closed convex hull of its preserved extreme points (see formal definition below), then given a norm-one Lipschitz function φ : N −→ M we have that C φ is an isometry if, and only if, for every pair of different points x, y ∈ M such that m x,y is a preserved extreme point, we can find a pair of sequences x n and y n in N so that φ(x n ) → x, φ(y n ) → y and…”

“…AcknowledgementsThe author is deeply grateful to Antonio Jiménez-Vargas for sending him the final version of[14]. He also thanks Rafael Chiclana, Luis C. García-Lirola and Colin Petitjean for their comments that improved the final version of the paper.…”