Inertia theorems for operator Lyapunov equations

We study operator Lyapunov equations in which the infinitesimal generator is not necessarily stable, but it satisfies a spectrum decomposition assumption and it has at most finitely many unstable eigenvalues. Under mild conditions, these have unique self-adjoint solutions. We give conditions under which the number of negative eigenvalues of this solution equals the number of unstable eigenvalues of the generator. An application to the bounded real lemma is treated