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

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 $$\mathbb {R}$$ 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.