Solvability of the Lyapunov equation for nonselfadjoint second-order differential operators with nonlocal boundary conditions
Let \(A\) be a closed linear operator with dense domain \(D(A)\) which acts in a Hilbert space \(H\). The following abstract Cauchy problem is considered: \[ u_t = Au(t),\qquad t>0;\qquad u(0)=u_0, \] where the operator \(A\) is a nonselfadjoint second-order differential operator with discrete spectrum and satisfies the ordinary condition guaranteeing unique solvability of the problem. For application it is important to find a simple method for calculating the points of the spectrum. The study of the spectrum of nonselfadjoint operators is complicated in view of difficulties arising in application of variational methods. These difficulties are overcome by employing and studying the Lyapunov equation \[ A^*U + UA = V, \] where \(A\) and \(A^*\) are given operators, \(A^*\) is adjoint to \(A\), \(V=V^*\), \(V>0\), and \(U\) is the sought selfadjoint operator. The theory of the Lyapunov equation was developed by M. G. Krein for bounded operators. This theory was generalized by V. S. Belonosov to the case of unbounded operators. The study was conducted for differential operators with splitting boundary conditions as well as for operators with nonsplitting boundary conditions which are generated by selfadjoint differential expressions. This article is devoted to the solvability of the Lyapunov equation in the important case of nonselfadjoint second-order differential operator with nonsplitting boundary conditions which is generated by a nonselfadjoint differential expression.
scientific article; zbMATH DE number 1285998