Into linear isometries between spaces of Lipschitz functions

Abstract

In this paper we state a Lipschitz version of a known Holsztynski's theorem on linear isometries of C(X)-spaces. Let Lip(X) be the Banach space of all scalar-valued Lipschitz functions f on a compact metric space X endowed with the norm ||f||=max{||f||_\infty, L(f)}, where L(f) is the Lipschitz constant of f. We prove that any linear isometry T from Lip(X) into Lip(Y) satisfying that L(T(1_X))<1 is essentially a weighted composition operator in the form Tf(y)=\tau(y)f(\varphi(y)) for all f in Lip(X) and y in Y_0, where Y_0 is a closed subset of Y, \varphi is a Lipschitz map from Y_0 onto X with L(\varphi) less or equal than max{1,diam(X)}, and \tau is a function in Lip(Y) with ||\tau||=1 and |\tau(y)|= 1 for all y in Y_0. We improve this representation in the case of onto linear isometries and we classify codimension 1 linear isometries in two types.