STABILITY Of ISOMETRIES ON HILBERT SPACES

Abstract

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

Keywords

• ($\varepsilon$, p)-isometry;
• isometry;
• Hilbert space

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References

1. Trans. Amer. Math. Soc. v.245 Stability of isomertries P. M. Gruber https://doi.org/10.2307/1998866
2. Proc. Amer. Math. Soc. v.72 On the stability of the linear mappings in Banach spaces Th. M. Rassias https://doi.org/10.1090/S0002-9939-1978-0507327-1
3. Proc. Amer. Math. Soc. v.89 Stability of isometries on Banach spaces J. Gevirtz https://doi.org/10.1090/S0002-9939-1983-0718987-6
4. Asterisque v.131 Non linear perturbation of isometries J. Lindenstrauss;A. Szankowski
5. Proc. Amer. Math. Soc. v.128 On the stability of approximately additive mappings Y.-H. Lee;K.-W. Jun https://doi.org/10.1090/S0002-9939-99-05156-4
6. C.R. Acad. Sci. Paris Ser v.194 Surle transformations isometriques d'espaces vectoriels norme's S. Mazur;S. Ulam
7. Proc. Amer. Math. Soc. v.124 Almost linearity of e-bi-Lipschitz maps between real Banach spaces K.-W. Jun;D.-W. Park https://doi.org/10.1090/S0002-9939-96-03267-4
8. J. Math. Anal. Appl. v.242 Generalized stability of isometries G. Dolinar https://doi.org/10.1006/jmaa.1999.6649
9. Math. Ann. v.303 On non linear perturbation of isometries M. Omladic;P. Semrl https://doi.org/10.1007/BF01461008