Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Phase-isometries on real normed spaces   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
$\mathcal{L}^{\infty}(\Gamma)$-type space or an $\ell^1(\Gamma)$-space with
$\Gamma$ being an index set.

32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
19
The categories of this paper are: eess
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
15
The categories of this paper are: nucl-ex
5
The categories of this paper are: gr-qc