Trace norm bounds for stable Lyapunov operators (Q1893101)

In the introduction the authors note that Lyapunov equations of the form \(A^TX + XA = - Q\) have been extensively studied because of their deep connections with dynamical systems, control theory, and a variety of other areas. The lemma in Section 1 illustrates one of the basic methods of this paper: first apply the trace to an integral representation of the Lyapunov solution, and then interchange the trace and integral operators. Next, reverse the order of the matrices inside the trace, and finally take the trace outside the integral. In discussing dual norms in Section 2, they follow the presentation of \textit{R. Horn} and \textit{C. R. Johnson} [Topics in matrix analysis (1991; Zbl 0729.15001)]. Dual norms arise naturally in the area of quantum mechanics in showing that any unitarily invariant matrix norm is associated with a symmetric gauge function and vice versa. They end this section with a look at the problem of bounding the Frobenius norm of the inverse Lyapunov operator \(\Phi\). The trace results of the previous sections find useful application in the problem of estimating the sensitivity of solutions to the Riccati equation \(0 = G + A^TX + XA - XFX\) in Section 3. In the previous section they show that the power method gives upper and lower bounds on the Frobenius norm of \(\Phi\). The same approach gives upper and lower bounds on the Frobenius norm of the operator \(\Pi (n) : = \Phi (XMX)\).