Article contents
WIGNER’S THEOREM IN
${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$-TYPE SPACES
Published online by Cambridge University Press: 04 December 2017
Abstract
We investigate surjective solutions of the functional equation $$\begin{eqnarray}\displaystyle \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}\quad (x,y\in X), & & \displaystyle \nonumber\end{eqnarray}$$
$f:X\rightarrow Y$ 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.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © 2017 Australian Mathematical Publishing Association Inc.
Footnotes
The authors are supported by the Natural Science Foundation of China, Grant Nos. 11371201, 11201337, 11201338 and 11301384.
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