Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 207-217, 2017

Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains

Mehmet Çelik, Yunus E. Zeytuncu

Received September 2, 2015.   First published February 24, 2017.

Mehmet Çelik, Texas A&M University-Commerce, Department of Mathematics, 1600 Education Dr., Binnion Hall Room 303A, Commerce, Texas, 75429-3011, USA, e-mail:; Yunus E. Zeytuncu, University of Michigan-Dearborn, Department of Mathematics and Statistics, 4901 Evergreen Road, 2014 CASL Building, Dearborn, Michigan, 48128, USA, e-mail:

Abstract: On complete pseudoconvex Reinhardt domains in $\mathbb{C}^2$, we show that there is no nonzero Hankel operator with anti-holomorphic symbol that is Hilbert-Schmidt. In the proof, we explicitly use the pseudoconvexity property of the domain. We also present two examples of unbounded non-pseudoconvex domains in $\mathbb{C}^2$ that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols. In the first example the Bergman space is finite dimensional. However, in the second example the Bergman space is infinite dimensional and the Hankel operator $H_{\bar{z}_1 \bar{z}_2}$ is Hilbert-Schmidt.

Keywords: canonical solution operator for $\overline{\partial}$-problem; Hankel operator; Hilbert-Schmidt operator

Classification (MSC 2010): 47B35, 32A36, 47B10

DOI: 10.21136/CMJ.2017.0471-15

Full text available as PDF.

  [1] J. Arazy: Boundedness and compactness of generalized Hankel operators on bounded symmetric domains. J. Funct. Anal. 137 (1996), 97-151. DOI 10.1006/jfan.1996.0042 | MR 1383014 | Zbl 0880.47015
  [2] J. Arazy, S. D. Fisher, J. Peetre: Hankel operators on weighted Bergman spaces. Am. J. Math. 110 (1988), 989-1053. DOI 10.2307/2374685 | MR 0970119 | Zbl 0669.47017
  [3] M. Çelik, Y. E. Zeytuncu: Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complex ellipsoids. Integral Equations Oper. Theory 76 (2013), 589-599. DOI 10.1007/s00020-013-2070-4 | MR 3073947 | Zbl 1288.47028
  [4] P. Harrington, A. Raich: Defining functions for unbounded $C^m$ domains. Rev. Mat. Iberoam. 29 (2013), 1405-1420. DOI 10.4171/RMI/762 | MR 3148609 | Zbl 1288.26008
  [5] P. S. Harrington, A. Raich: Sobolev spaces and elliptic theory on unbounded domains in $\mathbb R^n$. Adv. Diff. Equ. 19 (2014), 635-692. MR 3252898 | Zbl 1301.46015
  [6] S. G. Krantz, S.-Y. Li, R. Rochberg: The effect of boundary geometry on Hankel operators belonging to the trace ideals of Bergman spaces. Integral Equations Oper. Theory 28 (1997), 196-213. DOI 10.1007/BF01191818 | MR 1451501 | Zbl 0903.47019
  [7] T. Le: Hilbert-Schmidt Hankel operators over complete Reinhardt domains. Integral Equations Oper. Theory 78 (2014), 515-522. DOI 10.1007/s00020-013-2103-z | MR 3180876 | Zbl 1318.47047
  [8] H. Li: Schatten class Hankel operators on the Bergman spaces of strongly pseudoconvex domains. Proc. Am. Math. Soc. 119 (1993), 1211-1221. DOI 10.2307/2159984 | MR 1169879 | Zbl 0802.47022
  [9] M. M. Peloso: Hankel operators on weighted Bergman spaces on strongly pseudoconvex domains. Ill. J. Math. 38 (1994), 223-249. MR 1260841 | Zbl 0812.47023
  [10] J. R. Retherford: Hilbert space: Compact operators and the trace theorem. London Mathematical Society Student Texts 27, Cambridge University Press, Cambridge (1993). MR 1237405 | Zbl 0783.47031
  [11] G. Schneider: 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. (electronic only) 4 (2007), Artical No. 1, pages 7. MR 2326997 | Zbl 1220.47040
  [12] J. J. O. O. Wiegerinck: Domains with finite-dimensional Bergman space. Math. Z. 187 (1984), 559-562. DOI 10.1007/BF01174190 | MR 0760055 | Zbl 0534.32001
  [13] K. H. Zhu: Hilbert-Schmidt Hankel operators on the Bergman space. Proc. Am. Math. Soc. 109 (1990), 721-730. DOI 10.2307/2048212 | MR 1013987 | Zbl 0731.47028

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at
Subscribers of Springer need to access the articles on their site, which is

[Previous Article] [Next Article] [Contents of This Number] [Contents of Czechoslovak Mathematical Journal]