Plosker, SarahRamsey, Christopher2020-12-172022-05-312022-05-312019Plosker, 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.2018.09.003https://hdl.handle.net/20.500.14078/2103We 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).276.58KBPDFenAttribution-NonCommercial-NoDerivs (CC BY-NC-ND)Lyapunov Theoremoperator valued measurequantum probability measureatomic and nonatomic measuresArticle Post-Printhttps://doi.org/10.1016/j.jmaa.2018.09.003