Self-adjoint domains of products of differential expressions

Under the assumption that the product l(2) of the formally symmetric differential expression l of order n on [a, infinity) is partially separated in L-2[ a, infinity), we present a new characterization of self-adjoint boundary conditions for l(2). For two differential operators T-1(l) and T-2(l) associated with l, we show that the product T-2(l) T-1(l) is self-adjoint if and only if T-2(l) = T-1*(l). It extends the previous result in [1], where both T1(1) 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 l(2). (C) 2001 Academic Press.