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arXiv2026

Norm-Additive Maps on the Positive Unit Sphere of $C^1([0,1])$

Let $C^1(I)$ denote the space of all continuously differentiable real-valued functions on the closed unit interval $I=[0,1]$. For $1\le p\le\infty$, we equip $C^1(I)$ with the norm \[ \Vert{f}\Vert_{(p)} := \begin{cases} \left(|f(0)|^p+\Vert{f'}\Vert_{I}^p\right)^{1/p}, & 1\le p<\infty, \\ \max\{|f(0)|,\Vert{f'}\Vert_{I}\}, & p=\infty, \end{cases} \qquad (f\in C^1(I)), \] where $\Vert{\cdot}\Vert_I$ denotes the supremum norm on $I$. Let $\mathcal{S}_p^+:=\{f\in C^1(I):f(0)\ge0,\,f'\ge0,\,\Vert{f}\Vert_{(p)}=1\}$ be the positive unit sphere of $C^1(I)$. A map $T:\mathcal{S}_p^+\to \mathcal{S}_p^+$ is called norm-additive if \[ \|T(f)+T(g)\|_{(p)}=\|f+g\|_{(p)} \qquad (f,g\in \mathcal{S}_p^+). \] For $1 1$ is essential.

math.FA↗

arXiv2026

A variant of Tingley's problem for positive unit spheres of continuous functions that vanish at infinity

Let $S(C_0(X))^+$ and $S(C_0(Y))^+$ denote the positive parts of the unit spheres of $C_0(X)$ and $C_0(Y)$, where $X$ and $Y$ are locally compact Hausdorff spaces. We prove that every surjective isometry from $S(C_0(X))^+$ onto $S(C_0(Y))^+$ is a composition operator induced by a homeomorphism between $X$ and $Y$ . As a consequence, such a map extends to a surjective reallinear isometry from $C_0(X)$ onto $C_0(Y)$. We also characterize surjective phase-isometries on the positive unit sphere.

math.FA↗