Trace norm bounds for stable Kyapunov operators

Abstract

Certain aspects of stable Lyapunov operators can be easily studied by exploiting the linearity of the trace operator and its invariance under reversal of order in matrix products. For example, sharp upper and lower bounds on the trace of solutions to the stable Lyapunov equation can be obtained by applying the trace operator to a well-known integral 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 conditioning of solutions to the Riccati equation. In this regard, trace norm results can be obtained from well-known spectral norm results, since the trace and spectral norms are dual to each other. A somewhat deeper analysis involving the power method gives monotonically decreasing upper bounds on the Frobenius norms of these Lyapunov operators; these upper bounds complement the usual monotonically increasing lower bounds associated with the power method and provide a nice means of assessing the accuracy of the resulting Frobenius norm estimates.