Linear bijections preserving the Hölder seminorm

Let $(X,d)$ be a compact metric space and let $\alpha$ be a real number with $0<\alpha <1.$ The aim of this paper is to solve a linear preserver problem on the Banach algebra $C^{ {\alpha}}(X)$ of Hölder functions of order $\alpha$ from $X$ into $\mathbb{K}.$ We show that each linear bijection $T:C^{ {\alpha}} (X)\rightarrow C^{ {\alpha}}(X)$ having the property that $\alpha (T(f))=\alpha (f)$ for every $f\in C^{ {\alpha} }(X),$ where

$\alpha (f)=\sup \left\{ \frac{\left\vert f(x)-f(y)\right\vert }{d^{ {\alpha}} (x,y)}:x,y\in X, x\neq y\right\} ,$

is of the form $T(f)=\tau f\circ \varphi +\mu (f)1_X$ for every $f\in C^{ {\alpha} }(X),$ where $\tau \in \mathbb{K}$ with $\left\vert \tau \right\vert =1,$ $\varphi :X\rightarrow X$ is a surjective isometry and $\mu :C^{ {\alpha} }(X)\rightarrow \mathbb{K}$ is a linear functional.