Geodesic
Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 207-217
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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.
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.
DOI : 10.21136/CMJ.2017.0471-15
Classification : 32A36, 47B10, 47B35
Keywords: canonical solution operator for $\overline {\partial }$-problem; Hankel operator; Hilbert-Schmidt operator 
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Çelik, Mehmet; Zeytuncu, Yunus E. Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 207-217. doi: 10.21136/CMJ.2017.0471-15

[1] Arazy, J.: Boundedness and compactness of generalized Hankel operators on bounded symmetric domains. J. Funct. Anal. 137 (1996), 97-151. | DOI | MR | JFM

[2] Arazy, J., Fisher, S. D., Peetre, J.: Hankel operators on weighted Bergman spaces. Am. J. Math. 110 (1988), 989-1053. | DOI | MR | JFM

[3] Çelik, M., Zeytuncu, Y. E.: Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complex ellipsoids. Integral Equations Oper. Theory 76 (2013), 589-599. | DOI | MR | JFM

[4] Harrington, P., Raich, A.: Defining functions for unbounded $C^m$ domains. Rev. Mat. Iberoam. 29 (2013), 1405-1420. | DOI | MR | JFM

[5] Harrington, P. S., Raich, A.: Sobolev spaces and elliptic theory on unbounded domains in $\mathbb R^n$. Adv. Diff. Equ. 19 (2014), 635-692. | MR | JFM

[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 Equations Oper. Theory 28 (1997), 196-213. | DOI | MR | JFM

[7] Le, T.: Hilbert-Schmidt Hankel operators over complete Reinhardt domains. Integral Equations Oper. Theory 78 (2014), 515-522. | DOI | MR | JFM

[8] Li, H.: Schatten class Hankel operators on the Bergman spaces of strongly pseudoconvex domains. Proc. Am. Math. Soc. 119 (1993), 1211-1221. | DOI | MR | JFM

[9] Peloso, M. M.: Hankel operators on weighted Bergman spaces on strongly pseudoconvex domains. Ill. J. Math. 38 (1994), 223-249. | DOI | MR | JFM

[10] Retherford, J. R.: Hilbert space: Compact operators and the trace theorem. London Mathematical Society Student Texts 27, Cambridge University Press, Cambridge (1993). | MR | JFM

[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. (electronic only) 4 (2007), Artical No. 1, pages 7. | MR | JFM

[12] Wiegerinck, J. J. O. O.: Domains with finite-dimensional Bergman space. Math. Z. 187 (1984), 559-562. | DOI | MR | JFM

[13] Zhu, K. H.: Hilbert-Schmidt Hankel operators on the Bergman space. Proc. Am. Math. Soc. 109 (1990), 721-730. | DOI | MR | JFM

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