Hyers-Ulam stability of generalized $\varepsilon$-phase isometries

Abstract

In this paper, let $M$ and $N$ be two real Hilbert spaces. We first prove that the standard generalized $\varepsilon$-phase isometries have a wider range than the standard $\varepsilon$-phase isometries, and then we prove the stability of standard generalized $\varepsilon$-phase isometries. Finally,we further study the Hyers-Ulam stability of standard generalized $\varepsilon$-phase isometries. That is, we prove that there is a linear surjective isometry $U:M\rightarrow N$ and a phase function $\sigma:M\rightarrow \{-1,1\}$ such that
\begin{equation*}
\|f(u)-\sigma (u) U(u)\| \leq 2 \sqrt{2} \varepsilon, \forall u \in M
\end{equation*}
if $f: M \rightarrow N $ is a standard and almost surjective generalized $\varepsilon$-phase isometry.