Phase-isometries on real normed spaces

Abstract

We say that a mapping $f:X\to Y$ 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 ${\mathcal{L}}^{\infty }(\Gamma )$-type space or an ${\ell }^1(\Gamma )$-space with $\Gamma$ being an index set.

16pages

References in corpus (1)

• An elementary proof for the non-bijective version of Wigner's theorem