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.