Hilbert-Schmidt Hankel operators over complete Reinhardt domains
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 \textit{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 \textit{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.
scientific article; zbMATH DE number 6351693
scientific article