Quasicompact and Riesz composition operators on Banach spaces of Lipschitz functions on pointed metric spaces

Abstract

In this paper, we study quasicompact and Riesz composition operators on Banach spaces of Lipschitz–Hölder functions on pointed metric spaces. For a composition operator T on these spaces, we give an upper bound for \(r_e(T)\), the essential spectral radius of T, and establish a formula for \(r_e(T)\) whenever metric spaces are compact. We also give some necessary and some sufficient conditions that a composition operator T on these spaces to be quasicompact or Riesz. Finally, we get a relation for the set of eigenvalues and the spectrum of a quasicompact and Riesz composition operator on these spaces.

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References

  1. 1.

    Behrouzi, F.: Riesz and quasi-compact endomorphisms of Lipschitz algebras. Houston J. Math. 36(3), 793–802 (2010)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Dales, H.G.: Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, New Series, vol. 24. The Clarendon Press, Oxford (2000)

    Google Scholar 

  3. 3.

    Daneshmand, S., Alimohammadi, D.: Weighted composition operators between Lipschitz spaces on pointed metric spaces. Oper. Matrices 13(2), 545–561 (2019)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Feinstein, J.F., Kamowitz, H.: Quasicompact and Riesz endomorphisms of Banach algebras. J. Funct. Anal. 225, 427–438 (2005)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Feinstein, J.F., Kamowitz, H.: Quasicompact endomorphisms of commutative semiprime Banach algebras. Banach Center Publ. 91, 159–167 (2010)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Golbaharan, A., Mahyar, H.: Essential spectral radius of quasicompact endomorphisms of Lipschitz algebras. Rocky Mt. J. Math. 45(4), 1149–1164 (2015)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Golbaharan, A., Mahyar, H.: Weighted composition operators on Lipschitz algebras. Houston J. Math. 42(3), 905–917 (2016)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Jiménez-Vargas, A.: Norm-attaining composition operators on Lipschitz spaces. Taiwan. J. Math. 23(1), 129–144 (2019)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Jiménez-Vargas, A., Lacruz, M., Villegas-Vallecillos, M.: Essential norm of composition operators on Banach spaces of Hölder functions. Abstr. Appl. Anal. Article ID 590853 590853, 13 (2011)

    MATH  Google Scholar 

  10. 10.

    Jiménez-Vargas, A., Villegas-Vallecillos, M.: Compact composition operators on noncompact Lipschitz spaces. J. Math. Anal. Appl. 398, 221–229 (2013)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Kamowitz, H., Scheinberg, S.: Some properties of endomorphisms of Lipschitz algebras. Stud. Math. 96(3), 255–261 (1990)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Mayghani, M., Alimohammadi, D.: Quasicompact and Riesz composition endomorphisms of Lipschitz algebras of complex-valued bounded functions and their spectra. Bull. Iran. Math. Soc. 44, 531–558 (2018)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Mayghani, M., Alimohammadi, D.: Quasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions. Shahand Commun. Math. Anal. 9(1), 1–14 (2018)

    MATH  Google Scholar 

  14. 14.

    Sanatpour, A.H.: Quasicompact composition operators and power-contractive selfmaps. Annal. Funct. Anal. 7(2), 281–289 (2016)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Sherbert, D.R.: Banach algebras of Lipschitz functions. Pac. J. Math. 13, 1387–1399 (1963)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Weaver, N.: Lipschitz Algebras, 2nd edn. World Scientific, New Jersey (2018)

    Google Scholar 

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Acknowledgements

The authors would like to thank the referee for carefully reading the paper and his/her invaluable comments and suggestions. This research was in part supported by a grant from Arak University (no. 97/2321). The authors would like to thank this support.

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Correspondence to Davood Alimohammadi.

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Communicated by Martin Mathieu.

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Alimohammadi, D., Mayghani, M. Quasicompact and Riesz composition operators on Banach spaces of Lipschitz functions on pointed metric spaces. Adv. Oper. Theory (2020). https://doi.org/10.1007/s43036-020-00093-3

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Keywords

  • Essential norm
  • Essential spectral radius
  • Pointed metric space
  • Quasicompact operator
  • Riesz operator

Mathematics Subject Classification

  • 46J10
  • 47B48
  • 47B38