Norm-Additive Maps on the Positive Unit Sphere of C1([0,1])
Abstract
Let C1(I) denote the space of all continuously differentiable real-valued functions on the closed unit interval I=[0,1]. For 1≤ p≤∞, we equip C1(I) with the norm ‖f‖(p) := (|f(0)|p+‖f'‖Ip)1/p, & 1≤ p<∞, max{|f(0)|,‖f'‖I}, & p=∞, (f∈ C1(I)), where ‖·‖I denotes the supremum norm on I. Let 𝒮p+:={f∈ C1(I):f(0)≥0, f'≥0, ‖f‖(p)=1} be the positive unit sphere of C1(I). A map T:𝒮p+→ 𝒮p+ is called norm-additive if |T(f)+T(g)|(p)=|f+g|(p) (f,g∈ 𝒮p+). For 1<p≤∞, we prove that every surjective norm-additive map T:𝒮p+→𝒮p+ is of the form T(f)(t) = f(0)+∫0t f'(τ(s)) ds for all f∈𝒮p+ and t∈ I, where τ: I→ I is a homeomorphism. Conversely, every map of this form is a surjective norm-additive map. Consequently, every such map extends to a surjective linear isometry from (C1(I),‖·‖(p)) onto itself. For p=1, however, we construct a surjective norm-additive map on 𝒮1+ which is not of the above form and admits no linear extension to C1(I), showing that the assumption p>1 is essential.
Related
- The norm reveals projections in the positive unit sphere of an arbitrary C*-algebra
- Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries
- Plasticity of the unit sphere of the space c0
- The Second Gap for Minimal Hypersurfaces in the Unit Sphere with Constant Cubic Trace
- Orthogonal Pairs in Maps from the Sphere to the Circle
BibTeX
@misc{ezumi2026normadditive,
title = {{Norm-Additive Maps on the Positive Unit Sphere of $C^1([0,1])$}},
author = {Kazuki Ezumi and Takeshi Miura},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02862},
eprint = {2610.02862},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02862},
}
RIS
TY - UNPB
TI - Norm-Additive Maps on the Positive Unit Sphere of C^1([0,1])
AU - Kazuki Ezumi
AU - Takeshi Miura
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02862
UR - https://arxiv.org/abs/2610.02862
AB - Let C^1(I) denote the space of all continuously differentiable real-valued functions on the closed unit interval I=[0,1]. For 1≤ p≤∞, we equip C^1(I) with the norm ‖f‖_(p) := (|f(0)|^p+‖f'‖_I^p)^1/p, & 1≤ p<∞, max{|f(0)|,‖f'‖_I}, & p=∞, (f∈ C^1(I)), where ‖·‖_I denotes the supremum norm on I. Let 𝒮_p^+:={f∈ C^1(I):f(0)≥0, f'≥0, ‖f‖_(p)=1} be the positive unit sphere of C^1(I). A map T:𝒮_p^+→ 𝒮_p^+ is called norm-additive if |T(f)+T(g)|_(p)=|f+g|_(p) (f,g∈ 𝒮_p^+). For 1<p≤∞, we prove that every surjective norm-additive map T:𝒮_p^+→𝒮_p^+ is of the form T(f)(t) = f(0)+∫_0^t f'(τ(s)) ds for all f∈𝒮_p^+ and t∈ I, where τ: I→ I is a homeomorphism. Conversely, every map of this form is a surjective norm-additive map. Consequently, every such map extends to a surjective linear isometry from (C^1(I),‖·‖_(p)) onto itself. For p=1, however, we construct a surjective norm-additive map on 𝒮_1^+ which is not of the above form and admits no linear extension to C^1(I), showing that the assumption p>1 is essential.
ER -