Surjective isometries on function spaces with derivatives
M. G. Cabrera-Padilla, A. Jiménez-Vargas, Takeshi Miura, Moisés Villegas-Vallecillos
TL;DR
The article advances the theory of surjective isometries on function spaces endowed with derivative-based norms by providing a complete boundary-based description of all surjective isometries on $A$ where $\|f\|=\|f\|_X+\|d(f)\|_Y$ and $d:A\to B$ is surjective. The authors construct an isometric embedding of $A$ into a subspace of $C(\mathcal{Z})$ with $\mathcal{Z}=\partial A\times\partial B\times\mathbb{T}$, analyze the Choquet boundary, and relate isometries to boundary data via a Mazur–Ulam-induced real-linear isometry on the embedded algebra. They derive that any surjective isometry $T$ has the form $T(f)(x)-T(0)(x)=c\langle f(\phi(x))\rangle^{\varepsilon_0(x)}$ on $\partial A$ and $d(T(f)-T(0))(y)=u(y)\langle d(f)(\psi(y))\rangle^{\varepsilon_1(y)}$ on $\partial B$, where $c\in\mathbb{T}$, $\phi,\psi$ are boundary homeomorphisms, $u$ is continuous with values in $\mathbb{T}$, and $\varepsilon_0,\varepsilon_1$ are sign functions. The paper also provides a converse, a corollary for the one-point base space, and several illustrative examples, showing how the framework unifies previous results on isometries for spaces such as $C^1([0,1])$ and analytic function spaces.
Abstract
Let $A$ be a complex Banach space with a norm $\|f\|=\|f\|_X+\|d(f)\|_Y$ for $f\in A$, where $d$ is a complex linear map from $A$ onto a Banach space $B$, and $\|\cdot\|_K$ represents the supremum norm on a compact Hausdorff space $K$. In this paper, we characterize surjective isometries on $(A,\|\cdot\|)$, which may be nonlinear. This unifies former results on surjective isometries between specific function spaces.