Linear bijections preserving the Hölder seminorm

Abstract:

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 \begin{equation*} \alpha (f)=\sup \left \{ \frac {\left | f(x)-f(y)\right | }{d^{ {\alpha }} (x,y)}:x,y\in X,\ x\neq y\right \} , \end{equation*} 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 $\left | \tau \right | =1,$ $\varphi :X\rightarrow X$ is a surjective isometry and $\mu :C^{ {\alpha } }(X)\rightarrow \mathbb {K}$ is a linear functional.