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arXiv · 1607.06737

Cauchy transforms arising from homomorphic conditional expectations parametrize free Pick functions but those arising from conditional expectations do not

J. E. Pascoe·Ryan Tully-Doyle
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Abstract

Nevanlinna showed that Cauchy transforms of probability measures parametrize all functions from the upper half plane into itself satisfying a certain asymptotic condition at infinity. We show that the correspondence fails in general for the unbounded case for somewhat trivial reasons; however, we show that in a setting of "homomorphic" operator valued free probability that Cauchy transforms of homomorphic conditional expectations parametrize free Pick functions.

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BibTeXRIS

J. E. Pascoe, Ryan Tully-Doyle. 2016-07-22. Cauchy transforms arising from homomorphic conditional expectations parametrize free Pick functions but those arising from conditional expectations do not. https://arxiv.org/abs/1607.06737

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