An operator-valued Lyapunov theorem
Lyapunov Theorem, operator valued measure, quantum probability measure, atomic and nonatomic measures
We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak*-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).
Plosker, S., & Ramsey, C. (2019). An operator-valued Lyapunov theorem. Journal of Mathematical Analysis and Applications 469(1), 117-125. https://doi.org/10.1016/j.jmaa.2018.09.003
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