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THE WIGNER PROPERTY OF SMOOTH NORMED SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices Equations and inequalities involving nonlinear operators

Published online by Cambridge University Press:  09 May 2024

XUJIAN HUANG Open the ORCID record for XUJIAN HUANG [Opens in a new window] ,
JIABIN LIU Open the ORCID record for JIABIN LIU [Opens in a new window]  and
SHUMING WANG Open the ORCID record for SHUMING WANG [Opens in a new window]
XUJIAN HUANG
Affiliation:
Institute of Operations Research and Systems Engineering, College of Science, Tianjin University of Technology, Tianjin 300384, PR China e-mail: huangxujian86@sina.com
JIABIN LIU
Affiliation:
College of Science, Tianjin University of Technology, Tianjin 300384, PR China e-mail: liujb98@163.com
SHUMING WANG*
Affiliation:
College of Science, Tianjin University of Technology, Tianjin 300384, PR China
*
e-mail: smwang@email.tjut.edu.cn
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Abstract

We prove that every smooth complex normed space X has the Wigner property. That is, for any complex normed space Y and every surjective mapping $f: X\rightarrow Y$ satisfying

$$ \begin{align*} \{\|f(x)+\alpha f(y)\|: \alpha\in \mathbb{T}\}=\{\|x+\alpha y\|: \alpha\in \mathbb{T}\}, \quad x,y\in X, \end{align*} $$

where $\mathbb {T}$ is the unit circle of the complex plane, there exists a function $\sigma : X\rightarrow \mathbb {T}$ such that $\sigma \cdot f$ is a linear or anti-linear isometry. This is a variant of Wigner’s theorem for complex normed spaces.

Keywords

Wigner’s theoremWigner propertyphase-isometrysmooth space

MSC classification

Primary: 46B04: Isometric theory of Banach spaces
Secondary: 39B05: General 47J05: Equations involving nonlinear operators (general)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 9
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first and second authors were supported by the Natural Science Foundation of Tianjin Municipal Science and Technology Commission (Grant No. 22JCYBJC00420) and the National Natural Science Foundation of China (Grant No. 12271402). The third author was supported by the National Natural Science Foundation of China (Grant Nos. 12201459 and 12071358).

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