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Quasicompact and Riesz Composition Endomorphisms of Lipschitz Algebras of Complex-Valued Bounded Functions and Their Spectra

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Abstract

In this paper, we study quasicompact and Riesz composition endomorphisms of Lipschitz algebras of complex-valued bounded functions on metric spaces, not necessarily compact. We give some necessary and some sufficient conditions that a composition endomorphism of these algebras to be quasicompact or Riesz. We also establish an upper bound and a formula for the essential spectral radius of a composition endomorphism T of these algebras under some conditions which implies that T is quasicompact or Riesz. Finally, we get a relation for the set of eigenvalues and the spectrum of a quasicompact and Riesz endomorphism of these algebras.

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  • 16 May 2018

    In the original publication of this article, the e-mail address of the corresponding author was inadvertently entered incorrectly.

References

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

    MathSciNet  MATH  Google Scholar 

  2. Dales, H.G.: Banach algebras and automatic continuity. In: London Mathematical Society Monograph, vol. 24. The Clarendon Press, Oxford (2000)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  6. Golbaharan, A., Mahyar, H.: Linear operators of Banach spaces with range in Lipschitz algebras. Bull. Iran. Math. Soc. 42(1), 69–78 (2016)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Jiménez-Vargas, A., Lacruz, M., Villegas-Vallecillos, M. : Essential norm of composition operators on Banach spaces of Hölder functions. In: Abstract and Applied Analysis. Hindawi Publishing Corporation (2011) (article ID 590853)

  9. Kamowitz, H., Scheinberg, S.: Some properties of endomorphisms of Lipschitz algebras. Stud. Math. 24(3), 255–261 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mahyar, H., Sanatpour, A.H.: Quasicompact endomorphisms of Lipschitz algebras of analytic functions. Publ. Math. Debr. 76/1–2, 135–145 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Sherbert, D.R.: The structure of ideals and point derivations in Banach algebras of Lipschitz functions. Trans. Am. Math. Soc. 111, 240–272 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  14. Weaver, N.: Lipschitz Algebras. World Scientific, River Edge (1999)

    Book  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for her/his very insightful suggestions and comments, in particular, for calling their attention to the Lemma 2.8 of the reference [6], resulted in improving and generalizing the Theorems 2.2 and 2.22.

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Authors and Affiliations

  1. Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-3697, Tehran, Iran

    Maliheh Mayghani

  2. Department of Mathematics, Faculty of Science, Arak University, Arak, 38156-8-8349, Iran

    Davood Alimohammadi

Authors
  1. Maliheh Mayghani
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  2. Davood Alimohammadi
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Correspondence to Davood Alimohammadi.

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Communicated by Hamid Reza Ebrahimi Vishki.

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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). https://doi.org/10.1007/s41980-018-0021-1

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  • DOI: https://doi.org/10.1007/s41980-018-0021-1

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