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Hyers-Ulam stability of isometries and non-expansive maps between spaces of continuous functions


Author: Igor A. Vestfrid
Journal: Proc. Amer. Math. Soc. 145 (2017), 2481-2494
MSC (2010): Primary 46B04, 46E15; Secondary 41A65
DOI: https://doi.org/10.1090/proc/13383
Published electronically: February 10, 2017
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Abstract: We introduce a notion of an $ \varepsilon $-non-expansive map and study the problem of uniform approximation of such a map by a non-expansive map. We apply then obtained results to show the following Hyers-Ulam stability of $ \varepsilon $-isometries: Let $ X$ be a Hausdorff compact space and $ Y$ be a metric compact space. Let $ {F\colon C(X)\to C(Y)}$ be an $ \varepsilon $-isometry. Then there is an isometry $ {H\colon C(X) \to C(Y)}$ such that

$\displaystyle \Vert F(f) - H(f) \Vert \leq 5\varepsilon , \quad f\in C(X). $

If in addition for every proper closed subset $ S\subset Y$ there is an $ f\in C(X)$ with $ \vert F(f)(z)\vert<\Vert F(f)\Vert - 3.5\varepsilon $ for every $ z\in S$, then $ H$ can be chosen linear.

This assertion does not hold for the $ \ell _p$ norm with $ 1< p<\infty $.


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Additional Information

Igor A. Vestfrid
Affiliation: Nehemya Street, 21/6, 32294 Haifa, Israel
Email: igor.vestfrid@gmail.com

DOI: https://doi.org/10.1090/proc/13383
Keywords: Stability, non-surjective isometry, non-expansive map, Banach spaces of continuous functions
Received by editor(s): May 12, 2016
Received by editor(s) in revised form: July 9, 2016
Published electronically: February 10, 2017
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2017 American Mathematical Society