Hypercyclic Toeplitz Operators


We study hypercyclicity of the Toeplitz operators in the Hardy space \({H^{2}(\mathbb{D})}\) with symbols of the form \({p(\overline{z}) + \varphi(z)}\), where \({p}\) is a polynomial and \({\varphi \in H^{\infty}(\mathbb{D})}\). We find both necessary and sufficient conditions for hypercyclicity which almost coincide in the case when deg \({p =1}\).

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


  1. 1

    Bayart F., Matheron E.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  2. 2

    Bourdon P.S.: Density of the polynomials in Bergman spaces. Pac. J. Math. 130(2), 215–221 (1987)

    MathSciNet  Article  Google Scholar 

  3. 3

    Bourdon P.S., Shapiro J.H.: Hypercyclic operators that commute with the Bergman backward shift. Trans. Am. Math. Soc. 352, 5293–5316 (2000)

    MathSciNet  Article  Google Scholar 

  4. 4

    Caughran, J.G.: Polynomial approximation and spectral properties of composition operators on H 2. Indiana Univ. Math. J. 21(1), 81–84 (1971)

  5. 5

    Duren P.L.: Theory of H p Spaces. Academic Press, New-York (1970)

    Google Scholar 

  6. 6

    Godefroy G., Shapiro J.H.: Operators with dense, invariant cyclic vector manifolds. J. Funct. Anal. 98, 229–269 (1991)

    MathSciNet  Article  Google Scholar 

  7. 7

    Grosse-Erdmann, K.-G., Peris Manguillot, A.: Linear chaos. In: Universitext. Springer, London (2011)

  8. 8

    Koosis P.: Introduction to H p. Cambridge University Press, Cambridge (1980)

    Google Scholar 

  9. 9

    Sarason D.: Weak-star generators of H . Pac. J. Math. 17(3), 519–528 (1966)

    MathSciNet  Article  Google Scholar 

  10. 10

    Shkarin, S.: Orbits of coanalytic Toeplitz operators and weak hypercyclicity. arXiv:1210.3191

Download references

Author information



Corresponding author

Correspondence to Anton Baranov.

Additional information

The authors were supported by the Grant MD-5758.2015.1. A. Lishanskii was partially supported by JSC “Gazprom Neft” and by RFBR Grant 14-01-31163.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Baranov, A., Lishanskii, A. Hypercyclic Toeplitz Operators. Results. Math. 70, 337–347 (2016). https://doi.org/10.1007/s00025-016-0527-x

Download citation

Mathematics Subject Classification

  • 47A16
  • 47B35
  • 30H10


  • Hypercyclic operator
  • Toeplitz operator
  • univalent function