Skip to main content
Log in

The Weighted Composition Operators on the Large Weighted Bergman Spaces

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

Abstract

In this paper, we characterize bounded, compact or Schatten class weighted composition operators acting on Bergman spaces with the exponential type weights. Moreover, we give the proof of the necessary part for the boundedness of \(C_\phi \) on large weighted Bergman spaces given by Kriete and MacCluer (J Indiana Univ Math 1(3):755–788, 1992).

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.

Similar content being viewed by others

References

  1. Asserda, S., Hichame, A.: Pointwise estimate for the Bergman kernel of the weighted Bergman spaces with exponential type weights. C. R. Acad. Sci. Paris Ser. I 352, 13–16 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arroussi, H., Park, I., Pau, J.: Schatten class Toeplitz operators acting on large weighted Bergman spaces. Studia Math. 352, 203–221 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Chalendar, I., Gallardo-Gutiérrez, E.A., Partington, J.R.: Weighted composition operators on the Dirichlet space: boundedness and spectral properties. Math. Annalen. 363, 1265–1279 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Contreras, M.D., Hernandez-Diaz, A.G.: Weighted composition operators on Hardy spaces. J. Math. Anal. Appl 263, 224–233 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)

    MATH  Google Scholar 

  6. Cuckovic, Z., Zhao, R.: Weighted composition operators on the Bergman space. J. Lond. Math. Soc. 70, 499–511 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Halmos, P.: Measure Theory. Springer, Berlin (1974)

    MATH  Google Scholar 

  8. Kriete, T., MacCluer, B.: Composition operators on large weighted Bergman spaces. J. Indiana Univ. Math. 41(3), 755–788 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lin, P., Rochberg, R.: Hankel operators on the weighted Bergman spaces with exponential type weights. Integr. Equ. Oper. Theory 21, 460–483 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ohno, S., Zhao, R.: Weighted composition operators on the Bloch space. Bull. Austral. Math. Soc. 63, 177–185 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Oleinik, V.L.: Embedding theorems for weighted classes of harmonic and analytic functions. J. Soviet Math. 9, 228–243 (1978)

    Article  MATH  Google Scholar 

  12. Pau, J., Peláez, J.A.: Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights. J. Funct. Anal. 259, 2727–2756 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Peláez, J.A., Rättyä, J.: Trace class criteria for Toeplitz and composition operators on small Bergman spaces. Adv. Math. 293, 606–643 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shapiro, J.H.: The essential norm of a composition operator. Ann. Math. 125, 375–404 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sharma, A.K., Ueki, S.I.: Composition operators between weighted Bergman spaces with admissible Bekollé weights. Banach J. Math. Anal. 8, 64–88 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhu, K.: Operator Theory in Function Spaces. Marcel Dekker, New York (1990)

    MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank the referee for indicating various mistakes and giving helpful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Inyoung Park.

Additional information

Communicated by Scott McCullough.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Park, I. The Weighted Composition Operators on the Large Weighted Bergman Spaces. Complex Anal. Oper. Theory 13, 223–239 (2019). https://doi.org/10.1007/s11785-018-0768-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-018-0768-7

Keywords

Mathematics Subject Classification

Navigation