TRACE NORM BOUNDS FOR STABLE LYAPUNOV OPERATORS

Certain aspects of stable Lyapunov operators can be easily studied by exploiting the linearity of the trace operator and its invariance unde r reversal of order in matrix products. For example, sharp upper and l ower bounds on the trace of solutions to the stable Lyapunov equation can be obtained by applying the trace operator to a well-known integra l representation of these solutions. Other applications include using the connection between dual norms and the trace operator to obtain new results on the norms of Lyapunov operators associated with the condit ioning of solutions to the Riccati equation. In this regard, trace nor m results can be obtained from well-known spectral norm results, since the trace and spectral norms are dual to each other. A somewhat deepe r analysis involving the power method gives monotonically decreasing u pper bounds on the Frobenius norms of these Lyapunov operators; these upper bounds complement the usual monotonically increasing lower bound s associated with the power method and provide a nice means of assessi ng the accuracy of the resulting Frobenius norm estimates.