Skip to main content
Log in

Compact operators with BMO symbols on multiply-connected domains

  • Published:
Acta Scientiarum Mathematicarum Aims and scope Submit manuscript

Abstract

In this paper we study the problem of the boundedness and compactness of the Toeplitz operator Tφ on L2a(Ω), where Ω is a multiply-connected domain and φ is not bounded. We find a necessary and sufficient condition when the symbol is BMO. For this class we also show that the vanishing at the boundary of the Berezin transform is a necessary and sufficient condition for compactness. The same characterization is shown to hold when we analyze operators which are finite sums of finite products of Toeplitz operators with unbounded symbols.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Arazy, Membership of Hankel operators on planar domains in unitary ideals, Analysis at Urbana, Vol. 1, London Math. Soc. Lecture Notes Ser. 137, Cambridge University Press, 1989, 1–40.

    Google Scholar 

  2. J. Arazy, S. Fisher and J. Peetre, Hankel operators on planar domains, Constr. Approx., 6 (1990), 113–138.

    Article  MathSciNet  Google Scholar 

  3. S. Axler, Bergman spaces and their operators, Surveys of Some Recent Results in Operator Theory, Vol. 1, eds.: J. B. Conway and B. Morrel, Pitman Research Notes in Mathematics, 1988, 1–50.

    Google Scholar 

  4. S. Axler and D. Zheng, Compact operators via the Berezin transform, Indiana Univ. Math. J., 47 (1998), 387–400.

    Article  MathSciNet  Google Scholar 

  5. S. Bergman, The Kernel Function and the Conformal Mapping, Amer. Math. Soc. Math. Surveys 5, Amer. Math. Soc., 1950.

    Book  Google Scholar 

  6. R. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, 1972.

    MATH  Google Scholar 

  7. G.M. Goluzin, Geometric Theory of Functions of a Complex Variable, Trans. of Math. Monographs 26, Amer. Math. Soc., Providence, R. I., 1969.

    Book  Google Scholar 

  8. H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, Graduate Text in Math. 199, Springer-Verlag, 2000.

    Book  Google Scholar 

  9. H. Li, Hankel operators on the Bergman space of multiply-connected domains, J. Oper. Theory, 28 (1992), 321–335.

    MathSciNet  MATH  Google Scholar 

  10. M. Jovovic and D. Zheng, Compact operators and Toeplitz algebras on multiply-connected domains, J. Funct. Analysis, 261 (2011), 25–50.

    Article  MathSciNet  Google Scholar 

  11. J. Miao and D. Zheng, Compact operators on Bergman spaces, Integral Equations Operator Theory, 48 (2004), 61–79.

    Article  MathSciNet  Google Scholar 

  12. E. Nortdgren and P. Rosenthal, Boundary values of Berezin symbols, Operator Theory: Advances and Applications (Birkhäuser), 73 (1994), 362–368.

    MathSciNet  MATH  Google Scholar 

  13. R. Raimondo, Hilbert–Schmidt Hankel operators on the Bergman space of planar domains, Integral Equations Operator Theory, 57 (2006), 425–449.

    Article  MathSciNet  Google Scholar 

  14. N. Zorboska, Toeplitz operators with BMO symbols and the Berezin transform, Int. J. Math. M. Sc., 46 (2003), 2929–2945.

    Article  MathSciNet  Google Scholar 

  15. K. Zhu, Operator Theory in Function Spaces, Second Edition, Mathematical Surveys and Monographs 138, Amer. Math. Soc, Providence, RI, 2007.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Roberto Raimondo.

Additional information

Communicated by L. Kérchy

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raimondo, R. Compact operators with BMO symbols on multiply-connected domains. ActaSci.Math. 84, 643–658 (2018). https://doi.org/10.14232/actasm-017-283-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.14232/actasm-017-283-0

AMS Subject Classification

Key words and phrases

Navigation