Complex Analysis and Operator Theory

, Volume 13, Issue 1, pp 61–84 | Cite as

Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup

  • Serban Teodor BelinschiEmail author


In his article On the free convolution with a semicircular distribution, Biane found very useful characterizations of the boundary values of the imaginary part of the Cauchy–Stieltjes transform of the free additive convolution of a probability measure on \(\mathbb {R}\) with a Wigner (semicircular) distribution. Biane’s methods were recently extended by Huang (Int Math Res Not 12:4269–4292, 2015) to measures which belong to the partial free convolution semigroups introduced by Nica and Speicher. This note further extends some of Biane’s methods and results to free convolution powers of operator-valued distributions and to free convolutions with operator-valued semicirculars. In addition, it investigates properties of the Julia–Carathéodory derivative of the subordination functions associated to such semigroups, extending certain results from a previous article of H. Bercovici and the author to operator-valued maps.


Free noncommutative functions Free probability Operator-valued distributions Free convolution semigroups 

Mathematics Subject Classification

46L54 46E50 32H50 



I am very grateful to Hari Bercovici for having provided valuable feedback on an earlier version of this paper.


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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.CNRS - Institut de Mathématiques de ToulouseToulouseFrance

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