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.
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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