We use the notion of radial differential operator of order \(t>0\) to introduce the weighted composition–differentiation operator \(E^t_{\psi ,\varphi }(f)=\psi \cdot (R^t f)\circ \varphi \) on the Hardy and Bergman spaces of the unit ball and the polydisk in \(\mathbb {C}^n\). We obtain necessary and sufficient conditions on the functions \(\varphi \) and \(\psi \) to ensure that the operator \(E^t_{\psi ,\varphi }\) is Hilbert–Schmidt.

中文翻译：

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单位球上的希尔伯特-施密特组合-微分算子

我们使用\(t>0\)阶径向微分算子的概念来引入加权复合微分算子\(E^t_{\psi ,\varphi }(f)=\psi \cdot (R^tf) \circ \varphi \)上单位球的 Hardy 和 Bergman 空间以及\(\mathbb {C}^n\)中的多圆盘。我们在函数\(\varphi \)和\(\psi \)上获得充分必要条件，以确保算子\(E^t_{\psi ,\varphi }\)是 Hilbert–Schmidt。