Skip to main content
SpringerLink
Log in

Hilbert–Schmidt Hankel Operators with Anti-holomorphic Symbols on Complex Ellipsoids

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

On a complex ellipsoid in \({\mathbb{C}^n}\) , we show that there is no nonzero Hankel operator with an anti-holomorphic symbol that is Hilbert–Schmidt.

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. Arazy J., Fisher S.D., Peetre J.: Hankel operators on weighted Bergman spaces. Am. J. Math. 110(6), 989–1053 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arazy J.: Boundedness and compactness of generalized Hankel operators on bounded symmetric domains. J. Funct. Anal. 137(1), 97–151 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beatrous F., Li S.-Y.: On the boundedness and compactness of operators of Hankel type. J. Funct. Anal. 111(2), 350–379 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beatrous F., Li S.-Y.: Trace ideal criteria for operators of Hankel type. Ill. J. Math. 39(4), 723–754 (1995)

    MathSciNet  MATH  Google Scholar 

  5. Haslinger, F.: Compactness of the canonical solution operator to \({\overline{\partial}}\) restricted to Bergman spaces. In: Functional-Analytic and Complex Methods, Their Interactions, and Applications to Partial Differential Equations (Graz, 2001), pp. 394–400. World Sci. Publ., River Edge, NJ (2001)

  6. Krantz S.G., Li S.-Y., Rochberg R.: The effect of boundary geometry on Hankel operators belonging to the trace ideals of Bergman spaces. Integral Equ. Oper. Theory 28(2), 196–213 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Li H.: Schatten class Hankel operators on the Bergman spaces of strongly pseudoconvex domains. Proc. Am. Math. Soc. 119(4), 1211–1221 (1993)

    Article  MATH  Google Scholar 

  8. Park J.-D.: New formulas of the Bergman kernels for complex ellipsoids in \({{\mathbb{C}}^2}\). Proc. Am. Math. Soc. 136(12), 4211– (2008)

    Article  MATH  Google Scholar 

  9. Peloso M.M.: Hankel operators on weighted Bergman spaces on strongly pseudoconvex domains. Ill. J. Math. 38(2), 223–249 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Retherford, J.R.: Hilbert Space: Compact Operators and the Trace Theorem, London Mathematical Society Student Texts, vol. 27. Cambridge University Press, Cambridge (1993)

  11. Schneider, G.: A different proof for the non-existence of Hilbert–Schmidt Hankel operators with anti-holomorphic symbols on the Bergman space. Aust. J. Math. Anal. Appl. 4(2), Art. 1, 7 (2007)

    Google Scholar 

  12. Wiegerinck J.J.O.O.: Domains with finite-dimensional Bergman space. Math. Z. 187(4), 559–562 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhu K.H.: Hilbert–Schmidt Hankel operators on the Bergman space. Proc. Am. Math. Soc. 109(3), 721–730 (1990)

    Article  MATH  Google Scholar 

  14. Zhu, K.: Operator Theory in Function Spaces, Mathematical Surveys and Monographs, vol. 138, 2nd edn. American Mathematical Society, Providence, RI (2007)

Download references

Author information

Author notes

  1. Yunus E. Zeytuncu

    Present address: Department of Mathematics and Statistics, University of Michigan-Dearborn, Dearborn, MI, 48128, USA

Authors and Affiliations

  1. Department of Mathematics and Information Sciences, University of North Texas at Dallas, 7400 University Hills Blvd., Dallas, TX, 75241, USA

    Mehmet Çelik

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Yunus E. Zeytuncu

Authors
  1. Mehmet Çelik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunus E. Zeytuncu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunus E. Zeytuncu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Çelik, M., Zeytuncu, Y.E. Hilbert–Schmidt Hankel Operators with Anti-holomorphic Symbols on Complex Ellipsoids. Integr. Equ. Oper. Theory 76, 589–599 (2013). https://doi.org/10.1007/s00020-013-2070-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-013-2070-4

Mathematics Subject Classification (2010)

Keywords

Search

Navigation