Year: 2019
Type: preprint
Abstract: We say that a mapping $f: X \rightarrow Y$ between two real normed spaces is a phase-isometry if it satisfies the functional equation \begin{eqnarray*} \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \quad (x,y\in X).\end{eqnarray*} A generalized Mazur-Ulam question is whether every surjective phase-isometry is a multiplication of a linear isometry and a map with range $\{-1, 1\}$. This assertion is also an extension of a fundamental statement in the mathematical description of quantum mechanics, Wigner's theorem to real normed spaces. In this paper, we show that for every space $Y$ the problem is solved in positive way if $X$ is a smooth normed space, an $\mathcal{L}^{\infty}(Γ)$-type space or an $\ell^1(Γ)$-space with $Γ$ being an index set.
Source: arXiv (Cornell University)
Authors: Huang, Xujian, Tan, Dongni
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2943716497