Inertia theorems for operator Lyapunov inequalities

We study operator Lyapunov inequalities and equations for which the in1nitesimal generator is not necessarily stable, but it satis1es the spectrum decomposition assumption and it has at most 1nitely many unstable eigenvalues. Moreover, the input or output operators are not necessarily bounded, but are admissible. We prove an inertia result: under mild conditions, we show that the number ofunstable eigenvalues ofthe generator is less than or equal to the number ofnegative eigenvalues ofthe self -adjoint solution ofthe operator Lyapunov inequality. c 2001 Elsevier Science B.V. All rights reserved.