Complex Analysis and Operator Theory

, Volume 11, Issue 3, pp 651–674 | Cite as

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

  • Agbor Dieudonne


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.


Bounded symmetric domain m-Berezin transform Toeplitz operator 



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.


  1. 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)MathSciNetMATHGoogle Scholar
  2. 2.
    Axler, S., Zheng, D.: Compact operators via the Berezin transform. Indiana Univ. Math. J. 47, 387–400 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 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)MathSciNetCrossRefMATHGoogle Scholar
  4. 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)MathSciNetCrossRefMATHGoogle Scholar
  5. 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)MathSciNetCrossRefMATHGoogle Scholar
  6. 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 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)MathSciNetMATHGoogle Scholar
  8. 8.
    Berezin, F.A.: Covariant and contravariant symbols of operators. Math. USSR Izvest. 6(5), 1117–1151 (1972)CrossRefMATHGoogle Scholar
  9. 9.
    Choe, B.R., Lee, J.: Pluriharmonic symbols of commuting Toeplitz operators. Ill. J. Math. 37, 424–436 (1993)MathSciNetMATHGoogle Scholar
  10. 10.
    Englis, M.: Compact Toeplitz operators via the Berezin transform on bounded symmetric domains. Integr. Equ. Oper. Theory 33, 426–455 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Englis, M.: Erratun to “Compact Toeplitz operators via the Berezin transform on bounded symmetric domains”. Integr. Equ. Oper. Theory 34, 500–501 (1999)CrossRefGoogle Scholar
  12. 12.
    Englis, M.: A mean value theorem on bounded symmetric domains. Proc. Am. Math. Soc. 127(11), 127–131 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Faraut, J., Koranyi, A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 88, 3259–3268 (1990)MathSciNetCrossRefMATHGoogle Scholar
  14. 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)CrossRefMATHGoogle Scholar
  15. 15.
    Nam, K., Zheng, D., Zhong, C.: m-Berezin transform and compact operators. Rev. Mat. Iberoam. 22(3), 867–892 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Suárez, D.: Approximation and symbolic calculus for Toeplitz algebras on the Bergman space. Rev. Mat. Iberoam. 20, 563–610 (2004)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Suárez, D.: Approximation and the \(n\)-Berezin transform of operators on the Bergman space. J. Reine Angew. Math. 581, 175–192 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Upmeier, H.: Toeplitz operators and index theory in several complex variables, operator theory, advances and applications, vol. 81. Birkhäuser (1996)Google Scholar

Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of BueaBueaCameroon

Personalised recommendations