Complex symmetry of first-order differential operators on Hardy space

Given holomorphic functions $ψ_0$ and $ψ_1$, we consider first-order differential operators acting on Hardy space, generated by the formal differential expression $E(ψ_0,ψ_1)f(z)=ψ_0(z)f(z)+ψ_1(z)f'(z)$. 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.

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