  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.
