## Perturbations of isometries between Banach spaces.(English)Zbl 1250.46007

Suppose that $$E$$ and $$F$$ are two normed spaces, $$T$$ is a bijection and there is a function $$\mu:\mathbb{R}_{+} \to \mathbb{R}_{+}$$ satisfying the following conditions: (i) $$\|Tx-ty\|\leq \mu \|x-y\|$$, $$x, y \in E$$, (ii) $$\|T^{-1}u-T^{-1}v\| \leq \mu \|u-v\|$$, $$u,v\in F$$. Then $$T$$ is called a $$\mu$$-isometry.
In this paper, the author shows that if $$\frac{\mu (t)}{2} \leq {\mu (\frac{t}{2})}$$ for all $$t$$ and $$T:E \to F$$ is a $$\mu$$-isometry, then for each $$x,y \in E$$ and $$n \in \mathbb{Z}$$, $\left \|T(\frac{x+y}{2}) - \frac{Tx+Ty}{2} \right \| \leq \mu^{\circ(2^{n+1}-1)}(\frac{\|x-y\|}{2^{n+1}}),$ where $$\mu^{\circ n}=\mu \circ\dots \circ \mu$$ ($$n$$-fold composition).
As application, the author obtains a simple proof of a result of J. Gevirtz [Proc. Am. Math. Soc. 89, 633–636 (1983; Zbl 0561.46012)] answering the Hyers-Ulam problem and proves a non-linear generalization of the Banach-Stone theorem improving the results of K. Jarosz [Stud. Math. 93, No. 2, 97–107 (1989; Zbl 0695.46010)].

### MSC:

 46B04 Isometric theory of Banach spaces 46E40 Spaces of vector- and operator-valued functions 46B80 Nonlinear classification of Banach spaces; nonlinear quotients

### Citations:

Zbl 0561.46012; Zbl 0695.46010
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