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 $...
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| Published in: | Bulletin of the Australian Mathematical Society Vol. 97; no. 2; pp. 279 - 284 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cambridge, UK
Cambridge University Press
01.04.2018
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| Subjects: | |
| ISSN: | 0004-9727, 1755-1633 |
| Online Access: | Get full text |
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| Abstract | We investigate surjective solutions of the functional equation
$$\begin{eqnarray}\displaystyle \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}\quad (x,y\in X), & & \displaystyle \nonumber\end{eqnarray}$$
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. |
|---|---|
| AbstractList | 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. |
| Author | TAN, DONGNI JIA, WEIKE |
| Author_xml | – sequence: 1 givenname: WEIKE surname: JIA fullname: JIA, WEIKE email: jiaweike2017@163.com organization: Department of Mathematics, Tianjin University of Technology, 300384 Tianjin, China email jiaweike2017@163.com – sequence: 2 givenname: DONGNI orcidid: 0000-0002-0151-1203 surname: TAN fullname: TAN, DONGNI email: tandongni0608@sina.cn organization: Department of Mathematics, Tianjin University of Technology, 300384 Tianjin, China email tandongni0608@sina.cn |
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| Cites_doi | 10.1016/j.physleta.2008.09.052 10.1006/jfan.2002.3970 10.1016/j.physleta.2014.05.039 10.1016/S0034-4877(04)80012-0 10.1007/s00010-014-0296-0 10.1007/BF01818323 10.1017/S000497270900015X 10.1016/j.physleta.2013.08.017 10.5486/PMD.2012.5359 10.1016/0003-4916(90)90213-8 |
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| Copyright | 2017 Australian Mathematical Publishing Association Inc. |
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| DOI | 10.1017/S0004972717000910 |
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| Snippet | 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... |
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| Title | WIGNER’S THEOREM IN ${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$ -TYPE SPACES |
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