Advertisement

A note about the spectrum of composition operators induced by a rotation

  • 28 Accesses

Abstract

A characterization of those points of the unit circle which belong to the spectrum of a composition operator \(C_{\varphi }\), defined by a rotation \(\varphi (z)=rz\) with \(|r|=1\), on the space \(H_0(\mathbb {D})\) of all analytic functions which vanish at 0, is given. Examples show that the spectrum of \(C_{\varphi }\) need not be closed. In these examples the spectrum is dense but point 1 may or may not belong to it, and this is related to Diophantine approximation.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

References

  1. 1.

    Albanese, A.A., Bonet, J., Ricker, W.J.: Montel resolvents and uniformly mean ergodic semigroups of linear operators. Quaest. Math. 36, 253–290 (2013)

  2. 2.

    Arendt, W., Celariès, B., Chalendar, I.: In Koenigs’ footsteps: diagonalization of composition operators. J. Funct. Anal. 278, 108313 (2020)

  3. 3.

    Aron, R., Lindström, M.: Spectra of weighted composition operators on weighted Banach spaces of analytic functions. Israel J. Math. 141, 263–276 (2004)

  4. 4.

    Bonet, J., Domanski, P.: A note on mean ergodic composition operators on spaces of holomorphic functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 105, 389–396 (2011)

  5. 5.

    Carleson, L., Gamelin, T.W.: Complex Dynamics. Springer, New York (1993)

  6. 6.

    Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton, FL (1995)

  7. 7.

    Eisner, T., Farkas, B., Haase, M., Nagel, R.: Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics 272. Springer, New York (2015)

  8. 8.

    Eklund, T., Lindström, M., Mleczko, P., Rzeczkowski, M.: Spectra of weighted composition operators on abstract Hardy spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113, 267–279 (2019)

  9. 9.

    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008)

  10. 10.

    Ya, A.: Khinchin, Continued fractions. Translated from the third (1961) Russian edition. Reprint of the 1964 translation. Dover Publications, Inc., Mineola, NY (1997)

  11. 11.

    Meise, R., Vogt, D.: Introduction to Functional Analysis. The Clarendon Press Oxford University Press, New York (1997)

  12. 12.

    Queffélec, H., Queffélec, M.: Diophantine Approximation and Dirichlet series. Hindustain Book Agency, New Delhi (2013)

  13. 13.

    Shapiro, J.H.: Composition Operators and Classical Function Theory. Springer, New York (1993)

  14. 14.

    Shapiro, J.H.: Composition operators and Schröder’s functional equation. Contemporary Math. 213, 213–228 (1998)

  15. 15.

    Vasilescu, F.H.: Analytic functional calculus and spectral decompositions. Translated from the Romanian. Mathematics and its Applications (East European Series), 1. D. Reidel Publishing Co., Dordrecht (1982)

Download references

Acknowledgements

The author is grateful to J.L. Varona for providing him with useful references about number theory.

Author information

Correspondence to José Bonet.

Additional information

Publisher's Note

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

The research of this paper was partially supported by the projects MTM2016-76647-P and GV Prometeo/2017/102.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bonet, J. A note about the spectrum of composition operators induced by a rotation. RACSAM 114, 63 (2020). https://doi.org/10.1007/s13398-020-00788-5

Download citation

Keywords

  • Composition operator
  • Space of analytic functions
  • Rotation
  • Diophantine number

Mathematics Subject Classification

  • 47B33
  • 47A10
  • 46E10
  • 11K60