On the solvability of the Lyapunov equation for nonselfadjoint differential operators of order~\(2m\) with nonlocal boundary conditions (Q2759186)

Let \(A\) be a selfadjoint operator of order \(2m\) with nonlocal boundary conditions in spaces of vector-valued functions of one independent variable and let \(A^*\) be its adjoint. We assume that the spectra of the operators \(A^*\) and \(-A\) are disjoint and for some real \(\beta\) and \(\theta\) \((\pi/2<\theta<\pi)\) the resolvent \(R(A,\lambda)\) of \(A\) is uniformly bounded with respect to \(\lambda\) in the sector \(|\text{arg}(\lambda-\beta)|\leq \theta\). Under these assumptions the author investigates the solvability of the Lyapunov equation \(A^*U+UA=I\), where \(I\) is the identity operator and \(U\) denotes a selfadjoint operator to be found.