Linear bijections preserving the Hölder seminorm

Abstract

Let (X,d) be a compact metric space and let α be a real number with 0 < α < 1. The aim of this paper is to solve a linear preserver problem on the Banach algebra C^α(X) of Hölder functions of order α from X into K. We show that each linear bijection T : C^α(X) → C^α(X) having the property that α(T(f))=α(f) for every f ∈ C^α(X), where α(f)=sup{|f(x)-f(y)|/d(x, y)^α: x, y ∈ X, x \neq y}, is of the form T(f) = τf ◦ϕ+μ(f)1_X for every f ∈ C^α(X), where τ ∈ K with |τ| = 1, ϕ : X → X is a surjective isometry and μ : C^α(X) → K is a linear functional.