Bulletin of the
Korean Mathematical Society BKMS

Let $X$ and $Y$ be real Banach spaces and $\epsilon, p \ge 0$. A mapping $T$ between $X$ and $Y$ is called an $(\epsilon, p)$-isometry if $|\|T(x)-T(y)\|-\|x-y\| | \le \epsilon \|x-y\|^p$ for $x, y \in X$. Let $H$ be a real Hilbert space and $T:H \to H$ an $(\epsilon, p)$-isometry with $T(0)=0$. If $p \neq 1$ is a nonnegative number, then there exists a unique isometry $I:H\to H$ such that $\|T(x)-I(x)\| \le C(\epsilon)(\|x\|^{(1+p)/2}+\|x\|^p)$ for all $x \in H$, where $C(\epsilon)\to 0$ as $ \epsilon \to 0$.

Keywords: $(\epsilon,p)$-isometry, isometry, Hilbert spaces

MSC numbers: 46B20