MIN-PHASE-ISOMETRIES IN STRICTLY CONVEX NORMED SPACES

Abstract

AbstractSuppose 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 $\{-1,1\}$ .