Surjective isometries on function spaces with derivatives

Abstract

Let $A$ be a complex Banach space with a norm $\Vert f\Vert =\Vert f{\Vert }_X+\Vert d(f){\Vert }_Y$ for $f\in A$, where $d$ is a complex linear map from $A$ onto a Banach space $B$, and $\Vert \cdot {\Vert }_K$ represents the supremum norm on a compact Hausdorff space $K$. In this paper, we characterize surjective isometries on $(A,\Vert \cdot \Vert )$, which may be nonlinear. This unifies former results on surjective isometries between specific function spaces.