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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 
$\{-1,1\}$.
MAZUR–ULAM PROPERTY OF THE SUM OF TWO STRICTLY CONVEX BANACH SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 93 / Issue 3 / June 2016
- Published online by Cambridge University Press:
- 11 November 2015, pp. 473-485
- Print publication:
- June 2016
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In this article, we study the Mazur–Ulam property of the sum of two strictly convex Banach spaces. We give an equivalent form of the isometric extension problem and two equivalent conditions to decide whether all strictly convex Banach spaces admit the Mazur–Ulam property. We also find necessary and sufficient conditions under which the
$\ell ^{1}$-sum and the 
$\ell ^{\infty }$-sum of two strictly convex Banach spaces admit the Mazur–Ulam property.
