An operator-valued Lyapunov theorem
Functional Analysis
2018-09-12 v2 Quantum Physics
Abstract
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).
Cite
@article{arxiv.1801.03109,
title = {An operator-valued Lyapunov theorem},
author = {Sarah Plosker and Christopher Ramsey},
journal= {arXiv preprint arXiv:1801.03109},
year = {2018}
}
Comments
9 pages, added missing hypothesis to main theorem and minor changes, accepted to JMAA