Advertisement

Toeplitz Operators on Higher Cauchy–Riemann Spaces Over the Unit Ball

  • Lijia Ding
  • Kai Wang
Open Access
Article
  • 63 Downloads

Abstract

In this paper, we investigate some algebraic properties of Toeplitz operators over higher Cauchy–Riemann spaces \(C_{\alpha ,m}\) on the unit ball \(\mathbb {B}^d\) with \(d\ge 2\). We first discuss the Berezin transform on higher Cauchy–Riemann spaces. By making use of Berezin transform, we completely characterize (semi-)commuting Toeplitz operators with bounded pluriharmonic symbols over higher Cauchy–Riemann space \(C_{\alpha ,m}\). Moreover, we show that compact products of finite Toeplitz operators with a class of bounded pluriharmonic symbols only happen in the trivial case.

Keywords

Berezin transform \(\mathcal {M}\)-harmonic function Higher Cauchy–Riemann space Toeplitz operator Pluriharmonic function 

Mathematics Subject Classification

Primary 46E22 Secondary 47B35 

Notes

Acknowledgements

The authors would like to express their great gratitude to Professor K. Guo for his valuable guidance and encouragement over the years. The authors would also like to thank Professor G. Zhang for his helpful discussions and support.

References

  1. 1.
    Ahern, P., Flores, M., Rudin, W.: An invariant volume-mean-value property. J. Funct. Anal. 111, 380–397 (1993)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brown, A., Halmos, P.: Algebraic properties of Toeplitz operators. J. Reine Angew. Math. 213, 89–102 (1964)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Choe, B., Koo, H., Lee, J.: Toeplitz products with pluriharmonic symbols on the Hardy space over the ball. J. Math. Anal. Appl. 381, 365–382 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Engliš, M.: Functions invariant under the Berezin transform. J. Funct. Anal. 121, 233–254 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Engliš, M.: Compact Toeplitz operators via the Berezin transform on bounded symmetric domains. Integr. Equ. Oper. Theory 33, 426–455 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Engliš, M., Peetre, J.: Covariant Cauchy–Riemann operators and higher Laplacians on Kähler manifolds. J. Reine Angew. Math. 478, 17–56 (1996)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Engliš, M., Peetre, J.: Covariant differential operators and Green’s functions. Ann. Polon. Math. 66, 77–103 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Engliš, M., Zhang, G.: Toeplitz operators on higher Cauchy–Riemann spaces. Doc. Math. 22, 1081–1116 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Faraut, J., Korányi, A.: Analysis on Symmetric Cones. The Clarendon Press, New York (1994)zbMATHGoogle Scholar
  10. 10.
    Lee, J.: Properties of the Berezin transform of bounded functions. Bull. Aust. Math. Soc. 59, 21–31 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Loos, O.: Jordan Pairs, Lecture Notes in Mathematics, vol. 460. Springer, Berlin (1975)Google Scholar
  12. 12.
    Peetre, J., Peng, L., Zhang, G.: A Weighted Plancherel formula. I: The Case of the Disk, Applications to Hankel Operators. Technical Report, StockholmGoogle Scholar
  13. 13.
    Peetre, J., Zhang, G.: Invariant Cauchy–Riemann operators and relative discrete series of line bundles over the unit ball of \(\mathbb{C}^d\). Mich. Math. J. 45, 387–397 (1998)CrossRefGoogle Scholar
  14. 14.
    Rudin, W.: Function Theory in the Unit Ball of \(\mathbb{C}^n\). Grundlehren der Math, vol. 241. Springer, New York (1980)CrossRefGoogle Scholar
  15. 15.
    Shimeno, N.: The Plancherel formula for spherical functions with a one-dimensional K-type on a simply connected simple Lie group of Hermitian type. J. Funct. Anal. 121, 330–388 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Shimura, G.: On a class of nearly holomorphic automorphic forms. Ann. Math. 123, 347–406 (1986)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhang, G.: A weighted Plancherel formula. II: the case of the ball. Stud. Math. 102, 103–120 (1992)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhang, G.: Nearly holomorphic functions and relative discrete series of weighted \(L^2\)-spaces on bounded symmetric domains. J. Math. Kyoto Univ. 42, 207–221 (2002)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zheng, D.: Semi-commutators of Toeplitz operators on the Bergman space. Integr. Equ. Oper. Theory 25, 347–372 (1996)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zheng, D.: Commuting Toeplitz operators with pluriharmonic symbols. Trans. Am. Math. Soc. 350, 1595–1618 (1998)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zhu, K.: Operator Theory in Function Spaces. Operator Theory Advances & Applications, 2nd edn. American Mathematical Society, Providence (2007)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiP.R. China

Personalised recommendations