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Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains. (English) Zbl 06738513
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.

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 32A07 Special domains in $${\mathbb C}^n$$ (Reinhardt, Hartogs, circular, tube) (MSC2010) 32A36 Bergman spaces of functions in several complex variables 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.)
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