Representation of free Herglotz functions
arXiv:1607.00407 · doi:10.1512/iumj.2019.68.7730
Abstract
A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right halfplane. A classical Herglotz function has an integral representation against a positive measure on the unit circle. We prove a free analytic analogue of the Herglotz representation and describe how our representations specialize to the free probabilistic case. We also show that the set of representable Herglotz functions arising from noncommutative conditional expectations must be closed in a natural topology.
19 pages. To appear in Indiana University Mathematics Journal. Significant background added, especially with an eye toward the noncommutative probability context. Cumulative with the previous revision which expanded the results using the Ball-Marx-Vinnikov innovation in noncommutative Agler model theory
References in corpus (1)
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