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 prove that the essential norm of the composition operator Cφ induced by the symbol φ on the spaces lip0(X,dα) and Lip0(X,dα) is given by the formula ‖Cφ‖e=limt→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.