On the solvability of the Lyapunov equation for nonselfadjoint differential operators of order $2m$ with nonlocal boundary conditions
Volume 77 / 2001
Annales Polonici Mathematici 77 (2001), 79-104
MSC: 34G10, 34B05, 47E05, 47N20.
DOI: 10.4064/ap77-1-7
Abstract
This paper is devoted to the solvability of the Lyapunov equation $A^*U+UA=I$, where $A$ is a given nonselfadjoint differential operator of order $2m$ with nonlocal boundary conditions, $A^*$ is its adjoint, $I$ is the identity operator and $U$ is the selfadjoint operator to be found. We assume that the spectra of $A^*$ and $-A$ are disjoint. Under this restriction we prove the existence and uniqueness of the solution of the Lyapunov equation in the class of bounded operators.
