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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
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