Skip to main content
SpringerLink
Log in

m-Berezin Transform and Approximation of Operators on the Bergman Space Over Bounded Symmetric Domains

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

We study m-Berezin transforms of bounded operators on the Bergman space over a bounded symmetric domain, \(\Omega \). We use the m-Berezin transform to establish some results on norm approximation of bounded linear operators acting on the Bergman space by means of Toeplitz operators. We also use the m-Berezin transform to study compactness of bounded operators. In particular we show that a radial operator in the Toeplitz algebra is compact if and only if its Berezin transform vanishes on the boundary of the bounded symmetric domain.

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

Access this article

Log in via an institution

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. Agbor, D., Tchoundja, E.: Toeplitz operators with \(L^1\) symbols on Bergman spaces in the unit ball of \(\mathbb{C}^n\). Adv. Pure Appl. Math. 2(1), 65–88 (2011)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauer, W., Isralowitz, J.: Compactness characterization of operators in the Toeplitz algebra of the Fock space \(F^{p}_{\alpha }\). J. Funct. Anal. 263, 1323–1355 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bauer, W., Crispin, H.Y., Vasilevski, N.: \((m,\lambda )\)-Berezin transform and approximation of operators on weighted Bergman spaces over the unit ball. Oper. Theory Adv. Appl. 240, 45–68 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bauer, W., Crispin, H.Y., Vasilevski, N.: Eigenvalue characterization of radial operators on weighted Bergman spaces over the unit ball. Integr. Equ. Oper. Theory 78(2), 271–300 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Békollé, D., Berger, C.A., Coburn, L.A., Zhu, K.H.: BMO in the Bergman metric on bounded symmetric domains. J. Funct. Anal. 93, 310–350 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Békollé, D., Temgoua, K.: Reproducing kernels and \(L^{p}\) estimates for Bergman projections in Siegel domains of type II. Studia Math. 115, 219–239 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Berezin, F.A.: Covariant and contravariant symbols of operators. Math. USSR Izvest. 6(5), 1117–1151 (1972)

    Article  MATH  Google Scholar 

  9. Choe, B.R., Lee, J.: Pluriharmonic symbols of commuting Toeplitz operators. Ill. J. Math. 37, 424–436 (1993)

    MathSciNet  MATH  Google Scholar 

  10. Englis, M.: Compact Toeplitz operators via the Berezin transform on bounded symmetric domains. Integr. Equ. Oper. Theory 33, 426–455 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Englis, M.: Erratun to “Compact Toeplitz operators via the Berezin transform on bounded symmetric domains”. Integr. Equ. Oper. Theory 34, 500–501 (1999)

    Article  Google Scholar 

  12. Englis, M.: A mean value theorem on bounded symmetric domains. Proc. Am. Math. Soc. 127(11), 127–131 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Faraut, J., Koranyi, A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 88, 3259–3268 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mitkovski, M., Suárez, D., Wick, B.D.: The essential norm of operators on \(A^{p}_{\alpha }(\mathbb{B}_{n})\). Integr. Equ. Oper. Theory 75, 197–233 (2013)

    Article  MATH  Google Scholar 

  15. Nam, K., Zheng, D., Zhong, C.: m-Berezin transform and compact operators. Rev. Mat. Iberoam. 22(3), 867–892 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Suárez, D.: Approximation and symbolic calculus for Toeplitz algebras on the Bergman space. Rev. Mat. Iberoam. 20, 563–610 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Suárez, D.: Approximation and the \(n\)-Berezin transform of operators on the Bergman space. J. Reine Angew. Math. 581, 175–192 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Upmeier, H.: Toeplitz operators and index theory in several complex variables, operator theory, advances and applications, vol. 81. Birkhäuser (1996)

Download references

Acknowledgments

The author is grateful to Professor Wolfram Bauer for his useful comments and discussions and to Professor David Békollé for accepting to read through the manuscript and making some corrections.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Buea, P.O. Box 63, Buea, Cameroon

    Agbor Dieudonne

Authors
  1. Agbor Dieudonne
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Agbor Dieudonne.

Additional information

Communicated by Joseph Ball.

This work was partially supported by the Humboldt Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dieudonne, A. m-Berezin Transform and Approximation of Operators on the Bergman Space Over Bounded Symmetric Domains. Complex Anal. Oper. Theory 11, 651–674 (2017). https://doi.org/10.1007/s11785-015-0519-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-015-0519-y

Keywords

Search

Navigation