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MIN-PHASE-ISOMETRIES IN STRICTLY CONVEX NORMED SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 22 March 2023, pp. 152-160
- Print publication:
- February 2024
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- Article
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Suppose that X and Y are two real normed spaces. A map
$f:X\rightarrow Y$ is called a min-phase-isometry if it satisfies 
$$ \begin{align*} \min\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\min\{\|x+y\|,\|x-y\|\} \quad (x,y\in X). \end{align*} $$We present properties of min-phase-isometries in the case that Y is strictly convex and show that if a min-phase-isometry f (not necessarily surjective) fixes the origin, then it is phase-equivalent to a linear isometry, that is,
$f(x)=\varepsilon (x)g(x)$ for 
$x\in X$, where 
$g:X\rightarrow Y$ is a linear isometry and 
$\varepsilon $ is a map from X to 
$\{-1,1\}$.
