Hilbert–Schmidt composition–differentiation operators on the unit ball

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