Cauchy transforms arising from homomorphic conditional expectations parametrize noncommutative Pick functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:472

Article

Pascoe, J. E.¹  Tully-Doyle, Ryan² 

[1] Washington Univ, Dept Math, One Brookings Dr, St Louis, MO 63130 USA

[2] Univ New Haven, Dept Math & Phys, West Haven, CT 06516 USA

关键词: Free analysis;    Cauchy transforms;    Free probability;    Nevanlinna representation;    Noncommutative functions;   

DOI  :  10.1016/j.jmaa.2018.12.004

来源: Elsevier

【 摘 要 】

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 noncommutative Pick functions. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free