(m, λ)-Berezin Transform and Approximation of Operators on Weighted Bergman Spaces over the Unit Ball

  • Wolfram Bauer
  • Crispin Herrera Yañez
  • Nikolai Vasilevski
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 240)


We establish various results on norm approximations of bounded linear operators acting on the weighted Bergman space \( {\rm A}_\lambda ^2({\mathbb{B}^n}) \) over the unit ball by means of Toeplitz operators with bounded measurable symbols. The main tool here is the so-called (m, λ)-Berezin transform defined and studied in the paper. In a sense, this is a further development of the ideas and results of [6, 7, 9] to the case of operators acting on \( {\rm A}_\lambda ^2({\mathbb{B}^n}). \)


Toeplitz operator unit ball (m, λ)-Berezin transform norm approximation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    F.A. Berezin, Covariant and contravariant symbols of operators (Russian). Izv. Akad. Nauk. SSSR Ser. Mat. 36 (1972), 1134–1167.zbMATHMathSciNetGoogle Scholar
  2. [2]
    W. Bauer, C. Herrera Yañez, N. Vasilevski, Eigenvalue characterization of radial operators on weighted Bergman spaces over the unit ball, Integr. Equ. Oper. Theory, 78(2) (2014), 271–300.CrossRefGoogle Scholar
  3. [3]
    B.R. Choe, Y.J. Lee, Pluriharmonic symbols of commuting Toeplitz operators, Illinois J. Math. 37 (1993), 424–436.zbMATHMathSciNetGoogle Scholar
  4. [4]
    M. Engliš, Density of algebras generated by Toeplitz operators on Bergman spaces, Ark. Mat. 30 no. 2 (1992), 227–243.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    I.S. Gradshteyn, I.M. Ryzhik, Tables of integrals, series, and products, Academic Press, 1980.Google Scholar
  6. [6]
    M. Mitkovski, D. Suárez, and B.D. Wick, The essential norm of operators on \( A_{\alpha}^{p}(\mathbb{B}_{n}) \), Integr. Equ. Oper. Theory 75, no. 2 (2013), 197–233.CrossRefzbMATHGoogle Scholar
  7. [7]
    K. Nam, D. Zheng, and C. Zhong, m-Berezin transform and compact operators, Rev. Mat. Iberoamericana, 22(3) (2006), 867–892.Google Scholar
  8. [8]
    W. Rudin, Function theory in the unit ball of \( \mathbb{C}^{n} \), Fundamental principles of Mathematical Science 241, Springer-Verlag, New York-Berlin, 1980.Google Scholar
  9. [9]
    D. Suárez, Approximation and symbolic calculus for Toeplitz algebras on the Bergman space, Rev. Mat. Iberoamericana 20(2) (2004), 563–610.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    D. Suárez , The eigenvalues of limits of radial Toeplitz operators, Bull. London Math. Soc. v. 40 (2008), 631–641.CrossRefzbMATHGoogle Scholar
  11. [11]
    K. Zhu, Operator theory in function spaces, Marcel Dekker, Inc., 1990.Google Scholar
  12. [12]
    K. Zhu, Spaces of holomorphic functions in the unit ball, Springer-Verlag, 2005.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Wolfram Bauer
    • 1
  • Crispin Herrera Yañez
    • 2
  • Nikolai Vasilevski
    • 2
  1. 1.Mathematisches InstitutGeorg-August-UniversitätGöttingenGermany
  2. 2.Departamento de MatemáticasCINVESTAV del I.P.N.México D.F.México

Personalised recommendations