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

TL;DR

This paper extends boundary value characterizations of subordination functions in free convolution to operator-valued distributions and semicirculars, exploring their properties and derivatives within free probability theory.

Contribution

It introduces new results on subordination functions for operator-valued free convolutions, expanding previous scalar-based methods to operator-valued contexts.

Findings

Extended Biane's boundary value characterizations to operator-valued cases

Analyzed properties of Julia-Caratheodory derivatives of subordination functions

Provided new insights into free convolution powers with operator-valued semicircular distributions

Abstract

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 the real line with a Wigner (semicircular) distribution. Biane's methods were recently extended by Huang 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-Caratheodory derivative of the subordination functions associated to such semigroups, extending certain results from the article "Partially Defined Semigroups Relative to Multiplicative Free Convolution" by Bercovici and the author (reference [7] in the text).