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: mehmet.celik@tamuc.edu; Yunus E. Zeytuncu, University of Michigan-Dearborn, Department of Mathematics and Statistics, 4901 Evergreen Road, 2014 CASL Building, Dearborn, Michigan, 48128, USA, e-mail: zeytuncu@umich.edu

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.


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