数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (6): 2377-2386.doi: 10.1007/s10473-023-0603-8

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ISOMETRY AND PHASE-ISOMETRY OF NON-ARCHIMEDEAN NORMED SPACES*

Ruidong WANG, Wenting YAO   

  1. College of Science, Tianjin University of Technology, Tianjin 300384, China
  • 收稿日期:2022-06-28 修回日期:2023-05-22 发布日期:2023-12-08
  • 通讯作者: †Ruidong WANG, E-mail: wangruidong@tjut.edu.cn
  • 作者简介:Wenting YAO, E-mail: yaowenting103@163.com
  • 基金资助:
    Wang's research was supported by the Natural Science Foundation of China (12271402) and the Natural Science Foundation of Tianjin City (22JCYBJC00420).

ISOMETRY AND PHASE-ISOMETRY OF NON-ARCHIMEDEAN NORMED SPACES*

Ruidong WANG, Wenting YAO   

  1. College of Science, Tianjin University of Technology, Tianjin 300384, China
  • Received:2022-06-28 Revised:2023-05-22 Published:2023-12-08
  • Contact: †Ruidong WANG, E-mail: wangruidong@tjut.edu.cn
  • About author:Wenting YAO, E-mail: yaowenting103@163.com
  • Supported by:
    Wang's research was supported by the Natural Science Foundation of China (12271402) and the Natural Science Foundation of Tianjin City (22JCYBJC00420).

摘要: In this paper, we study isometries and phase-isometries of non-Archimedean normed spaces. We show that every isometry $f:S_{r}(X)\rightarrow S_{r}(X)$, where $X$ is a finite-dimensional non-Archimedean normed space and $S_{r}(X)$ is a sphere with radius $r\in \|X\|$, is surjective if and only if $\mathbb{K}$ is spherically complete and $k$ is finite. Moreover, we prove that if $X$ and $Y$ are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with $|2|=1$, any phase-isometry $f:X\rightarrow Y$ is phase equivalent to an isometric operator.

关键词: non-Archimedean normed spaces, isometry extension, Wigner's theorem

Abstract: In this paper, we study isometries and phase-isometries of non-Archimedean normed spaces. We show that every isometry $f:S_{r}(X)\rightarrow S_{r}(X)$, where $X$ is a finite-dimensional non-Archimedean normed space and $S_{r}(X)$ is a sphere with radius $r\in \|X\|$, is surjective if and only if $\mathbb{K}$ is spherically complete and $k$ is finite. Moreover, we prove that if $X$ and $Y$ are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with $|2|=1$, any phase-isometry $f:X\rightarrow Y$ is phase equivalent to an isometric operator.

Key words: non-Archimedean normed spaces, isometry extension, Wigner's theorem

中图分类号: 

  • 46B03