On approximate phase isometries

Abstract

A map $f:H\rightarrow K$, where H and K are real Hilbert spaces, is called an $\varepsilon$-phase isometry if $f(0)=0$ and

$$\begin{aligned} \min \{|\Vert f(x)-f(y)\Vert -\Vert x-y\Vert |,|\Vert f(x)-f(y)\Vert -\Vert x+y\Vert |\}\le \varepsilon , \end{aligned}$$

whenever $x,y\in H$. We prove the following stability property. If f is an $\varepsilon$-phase isometry, then there is a linear isometry V and a phase isometry $\gamma I$, where $\gamma :H\rightarrow \{-1,1\}$ is a phase mapping and I is the identity, such that $\Vert V^*f(x)-\gamma (x)x\Vert \le 2\sqrt{2}\varepsilon$ for all $x\in H$. If f is surjective, then V is surjective and $\Vert f(x)-\gamma (x)Vx\Vert \le 2\sqrt{2}\varepsilon$ for all $x\in H$.