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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:
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