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

  • Ruidong WANG ,
  • Wenting YAO
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  • College of Science, Tianjin University of Technology, Tianjin 300384, China
Wenting YAO, E-mail: yaowenting103@163.com

Received date: 2022-06-28

  Revised date: 2023-05-22

  Online published: 2023-12-08

Supported by

Wang's research was supported by the Natural Science Foundation of China (12271402) and the Natural Science Foundation of Tianjin City (22JCYBJC00420).

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.

Cite this article

Ruidong WANG , Wenting YAO . ISOMETRY AND PHASE-ISOMETRY OF NON-ARCHIMEDEAN NORMED SPACES*[J]. Acta mathematica scientia, Series B, 2023 , 43(6) : 2377 -2386 . DOI: 10.1007/s10473-023-0603-8

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