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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.

On the solvability of the Lyapunov equation for nonselfadjoint differential operators of order 2m with nonlocal boundary conditions