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Acta Mathematica Sinica, English Series

, Volume 35, Issue 2, pp 227–238 | Cite as

Radial Operators on the Weighted Bergman Spaces over the Polydisk

  • Ran LiEmail author
  • Yu Feng Lu
Article
  • 21 Downloads

Abstract

In this paper, we study radial operators in Toeplitz algebra on the weighted Bergman spaces over the polydisk by the (m, λ)-Berezin transform and find that a radial operator can be approximated in norm by Toeplitz operators without any conditions. We prove that the compactness of a radial operator is equivalent to the property of vanishing of its (0, λ)-Berezin transform on the boundary. In addition, we show that an operator S is radial if and only if its (m, λ)-Berezin transform is a separately radial function.

Keywords

Radial operators (m, λ)-Berezin transform weighted Bergman spaces Toeplitz operators 

MR(2010) Subject Classification

47B35 47B37 47A58 

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Notes

Acknowledgements

We thank the referees for their time and comments.

References

  1. [1]
    Bauer, W., Herrera Yañez, C., Vasilevski, N.: Eigenvalue characterization of radial operators on weighted Bergman spaces over the unit ball. Integr. Equ. Oper. Theory, 78, 271–300 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Bauer, W., Herrera Yañez, C., Vasilevski, N.: (m, λ)-Berezin transform and approximation of operators on weighted Bergman spaces over the Unit ball. Operator Theory: Advances and Applications, 240, 45–68 (2014)MathSciNetzbMATHGoogle Scholar
  3. [3]
    Conway, J. B.: A Course in Operator Theory, Graduate Studies in Mathematics, Volume 21, American Mathematical Society, Providence, RI, 2000zbMATHGoogle Scholar
  4. [4]
    Engliš, M.: Density of algebras generated by Toeplitz operators on Bergman spaces. Ark. Mat., 30(2), 227–243 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Engliš, M.: Compact Toeplitz operators via the Berezin transform on bounded symmetric domains. Integer. Equ. Oper. Theory, 33(4), 426–455 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Gradshteyn, I. S., Ryzhik, I. M.: Tables of Integrals, Series, and Products, Academic Press, 1980zbMATHGoogle Scholar
  7. [7]
    Li, R., Lu, Y. F.: (m, λ)-Berezin transform on the weighted Bergman spaces over the polydisk. J. Funct. Spaces, Art. ID 6804235, 11 pp (2016)Google Scholar
  8. [8]
    Mitkovski, M., Suárez, D., Wick, B. D.: The essential norm of operators on \(\mathcal{A}_\alpha^p(\mathbb{B}_n)\). Integr. Equ. Oper. Theory, 75(2), 197–233 (2013)CrossRefzbMATHGoogle Scholar
  9. [9]
    Nam, K., Zheng, D. C., Zhong, C.: m-Berezin transform and compact operators. Rev. Mat. Iberoamericana, 22, 867–892 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Nam, K., Zheng, D. C.: m-Berezin Transform on Polydisk. Integer. Equ. Oper. Theory, 56, 93–113 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Rudin, W.: Function Theory in Polydiscs, Mathematics Lecture Notes Series, W. A. Benjamin, Inc., New York, 1969zbMATHGoogle Scholar
  12. [12]
    Rudin, W.: Function Theory in the Unit Ball of ℂn, Springer-Verlag, Berlin, 1980CrossRefGoogle Scholar
  13. [13]
    Suárez, D.: Approximation and symbolic calculus for Toeplitz algebras on the Bergman space. Rev. Mat. Iberoamericana, 20, 563–610 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Zhu, K.: Operator Theory in Function Spaces, Marcel Dekker, Inc., New York, 1990zbMATHGoogle Scholar
  15. [15]
    Zhu, K.: Spaces of holomorphic functions in the unit ball, Graduate Studies in Mathematics, Volume 226, Springer-Verlag, New York, 2005Google Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science (CAS), Chinese Mathematical Society (CAS) and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesDalian University of TechnologyDalianP. R. China

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