Complex symmetry of first-order differential operators on Hardy space

Given holomorphic functions and , we consider first-order differential operators acting on Hardy space, generated by the formal differential expression . 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 -selfadjoint differential operators. The calculation of the point spectrum of some differential operators is performed in detail.