Given holomorphic functions $\psi_0$ and $\psi_1$, we consider first-order differential operators acting on Hardy space, generated by the formal differential expression $E(\psi_0,\psi_1)f(z)=\psi_0(z)f(z)+\psi_1(z)f'(z)$. We characterize these operators which are complex symmetric with respect to weighted composition conjugations. In parallel, as a basis of comparison, a characterization for differential operators which are hermitian is carried out. Especially, it is shown that hermitian differential operators are contained properly in the class of $\calC$-selfadjoint differential operators. The calculation of the point spectrum of some of these operators is performed in detail.

中文翻译：

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哈代空间上一阶微分算子的复对称性

给定全纯函数 $\psi_0$ 和 $\psi_1$，我们考虑作用于 Hardy 空间的一阶微分算子，由形式微分表达式 $E(\psi_0,\psi_1)f(z)=\psi_0(z) 生成f(z)+\psi_1(z)f'(z)$。我们描述了这些关于加权组合共轭的复杂对称的算子。同时，作为比较的基础，对厄密的微分算子进行了表征。特别是，证明了厄密微分算子正确地包含在 $\calC$-selfadjoint 微分算子类中。其中一些算子的点谱的计算是详细进行的。