Hölder seminorm preserving linear bijections and isometries

Abstract

Let (X, d) be a compact metric and 0 < α < 1. The space Lip α (X) of Holder functions of order α is the Banach space of all functions ƒ from X into \( \mathbb{K} \) such that ∥ƒ∥ = max{∥ƒ∥∞, L(ƒ)} < ∞, where $$ L(f) = sup\{ \left| {f(x) - f(y)} \right|/d^\alpha (x,y):x,y \in X, x \ne y\} $$ is the Holder seminorm of ƒ. The closed subspace of functions ƒ such that $$ \mathop {\lim }\limits_{d(x,y) \to 0} \left| {f(x) - f(y)} \right|/d^\alpha (x,y) = 0 $$ is denoted by lip α (X). We determine the form of all bijective linear maps from lip α (X) onto lip α (Y) that preserve the Holder seminorm.