Self-Adjoint Domains of Products of Differential Expressions

Under the assumption that the product l2 of the formally symmetric differential expression l of order n on [a, ∞) is partially separated in L2[a, ∞), we present a new characterization of self-adjoint boundary conditions for l2. For two differential operators T1(l) and T2(l) associated with l, we show that the product T2(l) T1(l) is self-adjoint if and only if T2(l)=T*1(l). It extends the previous result in [1], where both T1(l) and T2(l) are self-adjoint, singular limit-circle Sturm–Liouville operators. Furthermore, we also characterize the boundary conditions of the Friedrichs extension of the minimal operator generated by l2.