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Hilbert-Schmidt Hankel operators over complete Reinhardt domains. (English) Zbl 1318.47047
Given a bounded domain in $${\mathbb C}^n$$, the Bergman space $$A^2(\Omega)$$ over $$\Omega$$ is the Hilbert space consisting of all holomorphic functions in $$L^2(\Omega)$$, the Lebesgue space with respect to the Lebesgue volume measure over $$\Omega$$. Given $$\varphi\in L^2(\Omega)$$, the Hankel operator $$H_\varphi: A^2(\Omega)\to L^2(\Omega)\ominus A^2(\Omega)$$ is defined by $$H_\varphi h = (I-P)(\varphi h)$$ for all $$h\in A^2(\Omega)$$ for which $$\varphi h \in L^2(\Omega)$$.
In this paper, the author considers operators of the form $$H_{\overline f}$$, where $$f$$ is holomorphic and investigates when they are Hilbert-Schmidt. In the case of dimension $$1$$, it was proved by J. Arazy et al. [J. Reine Angew. Math. 406, 179–199 (1990; Zbl 0686.47023)] that $$H_{\overline f}$$ is Hilbert-Schmidt if and only if $$f'\in A^2(\Omega)$$. In higher dimensions $$(n \geq 2)$$, the situation becomes quite different. In fact, in the setting of the unit ball of $${\mathbb C}^n$$, it was proved by K. Zhu [Proc. Am. Math. Soc. 109, No. 3, 721–730 (1990; Zbl 0731.47028)] that $$H_{\overline f}$$ is Hilbert-Schmidt if and only if $$f$$ is constant. In this paper, the author extends Zhu’s result to the complete Reinhardt domains by means of a quite elementary approach. The author also indicates how his approach can be applied to certain general settings.

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 32A36 Bergman spaces of functions in several complex variables
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##### References:
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