Self-adjoint domains of products of differential expressions (Q5944082)

Let \(l\) be a formally symmetric differential expression on \([0,\infty)\), i.e., \(l=l^+\) where NEWLINE\[NEWLINEly=\sum^n_{j=0} p_j(t)y^{(j)}, \quad l^+y= \sum^n_{j=0} (-1)^j \bigl(\overline p_j (t)y\bigr)^{(j)}.NEWLINE\]NEWLINE Here \(p_j\in C^{n+j} [a,\infty)\) are complex-valued function \((j=0,\dots,n)\), \(p_n(t)\neq 0\) for \(t\in [a,\infty)\). We say that \(l^2(=l(ly))\) is partially separated in \(L^2[a,\infty)\) if \(y\in L^2[a,\infty)\), \(y^{(2n-1)}\in AC_{\text{loc}} [a,\infty)\) and \(l^2y\in L^2[a,\infty)\) together imply that \(ly\in L^2[a,\infty)\). Under the assumption that \(l^2\) is partially separated in \(L^2[a,\infty)\), the authors present a new characterization of self-adjoint boundary conditions for \(l^2\) and show that for two differential operators \(T_1(l)\) and \(T_2(1)\) associated with \(l\), the product \(T_2(l)T_1(L)\) is self-adjoint if and only if \(T_2(l)=T_1^*(l)\). Finally, they also characterize conditions that determine the Friedrichs extension of the minimal operator associated with \(l^2\).