In this paper, we study weighted composition-differentiation operators of the form \(D_{n,\varphi , \psi }\) on the Bergman space \(A^{2}\). We obtain explicit conditions for \(D_{n,\varphi , \psi }\) when it is complex symmetric with respect to conjugations \({\mathcal {C}}_{1}\) and \({\mathcal {C}}_{2}\) respectively. In addition, invariant subspaces of composition-differentiation operators are considered. Finally, we give spectral properties of weighted composition-differentiation operators.

中文翻译：

---

Bergman 空间上的加权组合微分算子

在本文中，我们研究了 Bergman 空间\(A^{2}\)上形式为\(D_{n,\varphi , \psi }\) 的加权组合微分算子。当\(D_{n,\varphi , \psi }\)关于共轭\({\mathcal {C}}_{1}\)和\({\mathcal { C}}_{2}\)分别。此外，还考虑了组合微分算子的不变子空间。最后，我们给出了加权成分微分算子的谱特性。