Skip to main content

Isometries between completely regular vector-valued function spaces

Periodica Mathematica Hungarica volume 84, pages 361–370 (2022)Cite this article

Abstract

In this paper, first we study surjective isometries (not necessarily linear) between completely regular subspaces A and B of \(C_0(X,E)\) and \(C_0(Y,F)\) where X and Y are locally compact Hausdorff spaces and E and F are normed spaces, not assumed to be either strictly convex or complete. We show that for a class of normed spaces F satisfying a newly defined property related to their T-sets, such an isometry is a (generalized) weighted composition operator up to a translation. Then we apply the result to study surjective isometries between A and B whenever A and B are equipped with certain norms rather than the supremum norm. Our results unify and generalize some recent results in this context.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1

References

  1. J. Araujo, L. Dubarbie, Noncompactness and noncompleteness in isometries of Lipschitz spaces. J. Math. Anal. Appl. 377, 15–29 (2011)

    MathSciNet  Article  Google Scholar 

  2. F. Botelho, R.J. Fleming, J. Jamison, Extreme points and isometries on vector-valued Lipschitz spaces. J. Math. Anal. Appl. 381, 821–832 (2011)

    MathSciNet  Article  Google Scholar 

  3. F. Botelho, J. Jamison, Surjective isometries on spaces of differentiable vector-valued functions. Stud. Math. 192, 39–50 (2009)

    MathSciNet  Article  Google Scholar 

  4. B. Cengiz, On extremely regular function spaces. Pac. J. Math. 49, 335–338 (1973)

    MathSciNet  Article  Google Scholar 

  5. K. de Leeuw, Banach spaces of Lipschitz functions. Stud. Math. 21, 55–66 (1962)

    MathSciNet  Article  Google Scholar 

  6. R. Fleming, J. Jamison, Isometries in Banach Spaces: Function Spaces, vol. 1 (CRC Press, Boca Raton, 2003)

    MATH  Google Scholar 

  7. R. Fleming, J. Jamison, Isometries in Banach Spaces: Vector-Valued Function Spaces and Operator Spaces, vol. 2 (CRC Press, Boca Raton, 2007)

    Book  Google Scholar 

  8. J. Font, Linear isometries between certain subspaces of continuous vector-valued functions. Ill. J. Math. 42, 389–397 (1998)

    MathSciNet  MATH  Google Scholar 

  9. O. Hatori, Hermitian operators and isometries on Banach algebras of continuous maps with values in a unital commutative \(C^*\)-algebras. J. Funct. Space 4, 1–14 (2018)

    MathSciNet  MATH  Google Scholar 

  10. M. Hosseini, Isometries on spaces of absolutely continuous vector-valued functions. J. Math. Anal. Appl. 463(1), 386–397 (2018)

    MathSciNet  Article  Google Scholar 

  11. A. Jamshidi, F. Sady, A note on nonlinear isometries between vector-valued function spaces. Linear Multilinear Algebra 68(2), 293–303 (2020)

    MathSciNet  Article  Google Scholar 

  12. M. Jerison, The space of bounded maps into a Banach space. Ann. Math. 52, 309–327 (1950)

    MathSciNet  Article  Google Scholar 

  13. A. Jiménez-Vargas, M. Villegas-Vallecillos, Into linear isometries between spaces of Lipschitz functions. Houst. J. Math. 34, 1165–1184 (2008)

    MathSciNet  MATH  Google Scholar 

  14. A. Jiménez-Vargas, M. Villegas-Vallecillos, Linear isometries between spaces of vector-valued Lipschitz functions. Proc. Am. Math. Soc. 137, 1381–1388 (2009)

    MathSciNet  Article  Google Scholar 

  15. K. Kawamura, H. Koshimizu, T. Miura, Norms on \(C^1([0,1])\) and their isometries. Acta Sci. Math. (Szeged) 84, 239–261 (2018)

    MathSciNet  Article  Google Scholar 

  16. K.S. Lau, A representation theorem for isometries of \(C(X, E)\). Pac. J. Math. 60, 229–233 (1975)

    MathSciNet  Article  Google Scholar 

  17. M. Mojahedi, F. Sady, Isometries on certain non-complete vector-valued function spaces. Acta. Sci. Math. (Szeged) 85, 613–627 (2019)

    MathSciNet  Article  Google Scholar 

  18. A. Ranjbar-Motlagh, A note on isometries of Lipschitz spaces. J. Math. Anal. Appl. 411, 555–558 (2014)

    MathSciNet  Article  Google Scholar 

  19. N.V. Rao, A.K. Roy, Linear isometries of some function spaces. Pac. J. Math. 38, 177–192 (1971)

    MathSciNet  Article  Google Scholar 

  20. A.K. Roy, Extreme points and linear isometries of the Banach space of Lipschitz functions. Can. J. Math. 20, 1150–1164 (1968)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, 14115-134, Iran

    Mojtaba Mojahedi & Fereshteh Sady

Authors
  1. Mojtaba Mojahedi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fereshteh Sady
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fereshteh Sady.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mojahedi, M., Sady, F. Isometries between completely regular vector-valued function spaces. Period Math Hung 84, 361–370 (2022). https://doi.org/10.1007/s10998-021-00413-x

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-021-00413-x

Keywords

  • Real-linear isometries
  • Vector-valued function spaces
  • T-sets
  • Strictly convex

Mathematics Subject Classification

  • Primary 47B38
  • 47B33
  • Secondary 46J10