Stability of isometries on Hilbert spaces
Bull. Korean Math. Soc. 2002 Vol. 39, No. 1, 141-151
Published online March 1, 2002
Kil-Woung Jun and Dal-Won Park
Chungnam National University, Kongju National University
Abstract : 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
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