Akemann and Weaver have shown an interesting variant of classical Lyapunov's theorem on ranges of non-atomic vector-valued measures in the setting of discrete frames. Their result is an extension of Weaver's conjecture, which was proved by Marcus, Spielman, and Srivastava in their breakthrough solution of the Kadison-Singer problem. Akemann and Weaver have conjectured a generalization of their Lyapunov-type theorem for higher rank operators. We show the validity of this conjecture for trace class operators by generalizing the main result of Marcus-Spielman-Srivastava to positive definite matrices of higher ranks. The method of the proof requires developing some new properties of mixed characteristic polynomials.
Copyright. All Rights Reserved.