An operator-valued Lyapunov theorem

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:469

Article

Plosker, Sarah¹  Ramsey, Christopher^1,2 

[1] Brandon Univ, Dept Math & Comp Sci, Brandon, MB R7A 6A9, Canada

[2] MacEwan Univ, Dept Math & Stat, Edmonton, AB T5J 4S2, Canada

关键词: Operator valued measure;    Quantum probability measure;    Atomic and nonatomic measures;    Lyapunov Theorem;   

DOI  :  10.1016/j.jmaa.2018.09.003

来源: Elsevier

【 摘 要 】

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). (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free