# Article

Full entry | PDF   (0.2 MB)
Keywords:
canonical solution operator for $\overline {\partial }$-problem; Hankel operator; Hilbert-Schmidt operator
Summary:
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.
References:
[1] Arazy, J.: 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] Arazy, J., Fisher, S. D., Peetre, J.: Hankel operators on weighted Bergman spaces. Am. J. Math. 110 (1988), 989-1053. DOI 10.2307/2374685 | MR 0970119 | Zbl 0669.47017
[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 10.1007/s00020-013-2070-4 | MR 3073947 | Zbl 1288.47028
[4] Harrington, P., Raich, A.: 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] 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 3252898 | Zbl 1301.46015
[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 10.1007/BF01191818 | MR 1451501 | Zbl 0903.47019
[7] Le, T.: 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] Li, H.: 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] Peloso, M. M.: Hankel operators on weighted Bergman spaces on strongly pseudoconvex domains. Ill. J. Math. 38 (1994), 223-249. MR 1260841 | Zbl 0812.47023
[10] Retherford, J. R.: 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] 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 2326997 | Zbl 1220.47040
[12] Wiegerinck, J. J. O. O.: Domains with finite-dimensional Bergman space. Math. Z. 187 (1984), 559-562. DOI 10.1007/BF01174190 | MR 0760055 | Zbl 0534.32001
[13] Zhu, K. H.: 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

Partner of