Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $.


References [Enhancements On Off] (What's this?)

  • [B] Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673
  • [CCTZ] Lixin Cheng, Qingjin Cheng, Kun Tu, and Jichao Zhang, A universal theorem for stability of $ \varepsilon $-isometries of Banach spaces, J. Funct. Anal. 269 (2015), no. 1, 199-214. MR 3345607, https://doi.org/10.1016/j.jfa.2015.04.015
  • [CZZ] Yu Zhou, Zihou Zhang, and Chunyan Liu, On linear isometries and $ \varepsilon $-isometries between Banach spaces, J. Math. Anal. Appl. 435 (2016), no. 1, 754-764. MR 3423426, https://doi.org/10.1016/j.jmaa.2015.10.035
  • [G] Julian Gevirtz, Stability of isometries on Banach spaces, Proc. Amer. Math. Soc. 89 (1983), no. 4, 633-636. MR 718987, https://doi.org/10.2307/2044596
  • [FSV] Tadeusz Figiel, Peter Šemrl, and Jussi Väisälä, Isometries of normed spaces, Colloq. Math. 92 (2002), no. 1, 153-154. MR 1899245, https://doi.org/10.4064/cm92-1-13
  • [H] W. Holsztyński, Lattices with real numbers as additive operators, Dissertationes Math. Rozprawy Mat. 62 (1969), 86. MR 0269560
  • [HU] D. H. Hyers and S. M. Ulam, On approximate isometries, Bull. Amer. Math. Soc. 51 (1945), 288-292. MR 0013219
  • [L] Gun-Marie Lövblom, Isometries and almost isometries between spaces of continuous functions, Israel J. Math. 56 (1986), no. 2, 143-159. MR 880288, https://doi.org/10.1007/BF02766121
  • [MU] S. Mazur and S. Ulam, Sur les transformations isométriques d'espaces vectoriels normés, Comp. Rend. Paris 194 (1932), 946-948.
  • [OŠ] Matjaž Omladič and Peter Šemrl, On nonlinear perturbations of isometries, Math. Ann. 303 (1995), no. 4, 617-628. MR 1359952, https://doi.org/10.1007/BF01461008
  • [Š] Peter Šemrl, Hyers-Ulam stability of isometries, Houston J. Math. 24 (1998), no. 4, 699-706. MR 1686636
  • [Se] Zbigniew Semadeni, Banach spaces of continuous functions. Vol. I, PWN--Polish Scientific Publishers, Warsaw, 1971. Monografie Matematyczne, Tom 55. MR 0296671
  • [Ve1] I. A. Vestfrid, Almost surjective $ \epsilon $-isometries of Banach spaces, Colloq. Math. 100 (2004), no. 1, 17-22. MR 2079344, https://doi.org/10.4064/cm100-1-3
  • [Ve2] Igor A. Vestfrid, Stability of almost surjective $ \varepsilon $-isometries of Banach spaces, J. Funct. Anal. 269 (2015), no. 7, 2165-2170. MR 3378871, https://doi.org/10.1016/j.jfa.2015.04.009

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46B04, 46E15, 41A65

Retrieve articles in all journals with MSC (2010): 46B04, 46E15, 41A65


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

American Mathematical Society