Essential norm of composition operators on Banach spaces of Hölder functions

Abstract

Let (X, d) be a pointed compact metric space, let 0 < α < 1, and let ϕ : X → X be a base point-preserving Lipschitz map. We show that the essential norm of the composition operator Cϕ induced by the symbol ϕ on the Lipschitz spaces lip0 (X, dα) and Lip0(X, dα) is given by the formula kCϕke = lim t→0 sup 0<d(x,y)<t d(ϕ(x), ϕ(y))α d(x, y) α whenever the dual space lip0 (X, dα)∗ has the approximation property. This happens in particular when X is an infinite compact subset of a finite-dimensional normed linear space.