Inertia theorems for operator Lyapunov inequalities

We study operator Lyapunov inequalities and equations for which the infinit esimal generator is not necessarily stable, but it satisfies the spectrum d ecomposition assumption and it has at most finitely many unstable eigenvalu es. Moreover, the input or output operators are not necessarily bounded, bu t are admissible. We prove an inertia result: under mild conditions, we sho w that the number of unstable eigenvalues of the generator is less than or equal to the number of negative eigenvalues of the self-adjoint solution of the operator Lyapunov inequality. (C) 2001 Elsevier Science B.V. All right s reserved.