WIGNER’S THEOREM IN ${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$ -TYPE SPACES

We investigate surjective solutions of the functional equation $$\begin{eqnarray}\displaystyle \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}\quad (x,y\in X), & & \displaystyle \nonumber\end{eqnarray}$$ where $f:X\rightarrow Y$ is a map between two real $...

Full description

Saved in:
Bibliographic Details
Published in:Bulletin of the Australian Mathematical Society Vol. 97; no. 2; pp. 279 - 284
Main Authors: JIA, WEIKE, TAN, DONGNI
Format: Journal Article
Language:English
Published: Cambridge, UK Cambridge University Press 01.04.2018
Subjects:
Mathematical functions
L ∞ -type spaces
Wigner’s theorem
primary 46B03
phase equivalent
secondary 46B04
ISSN:0004-9727, 1755-1633
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We investigate surjective solutions of the functional equation $$\begin{eqnarray}\displaystyle \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}\quad (x,y\in X), & & \displaystyle \nonumber\end{eqnarray}$$ where $f:X\rightarrow Y$ is a map between two real ${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$ -type spaces. We show that all such solutions are phase equivalent to real linear isometries. This can be considered as an extension of Wigner’s theorem on symmetry for real ${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$ -type spaces.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972717000910