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

Abstract

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 R 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-Carathéodory derivative of the subordination functions associated to such semigroups, extending results from [7].

Fractional free convolution powers

“…Proposition 1.2 was first established for sufficiently large k by Bercovici and Voiculescu [9], and then for all k ≥ 1 by Nica and Speicher [22]; a complex analysis proof using subordination was given by Belinschi-Bercovici [6,7] and Huang [16]. See also the recent paper [5] for further study of the subordination functions associated to these measures, and [16], [37] for further regularity and support properties of the µ ⊞k , and [2], [25] for an extension to the case when k is a completely positive map and µ takes values in a C * -algebra.…”

Outlier eigenvalues for non-Hermitian polynomials in independent i.i.d. matrices and deterministic matrices

“…It follows from [4] that the support in M sa m (C) of the addition of a semicircular s of variance η and a selfadjoint noncommutative random variable y ∈ (M m (A), id m ⊗ ϕ) which is free with amalgamation over M m (C) with s, is given via its complement in terms of y and the functions…”

Outlier eigenvalues for non-Hermitian polynomials in independent i.i.d. matrices and deterministic matrices

“…It follows from [4] that the support in M m (C) sa of the addition of a semicircular s of variance η and a selfadjoint noncommutative random variable y ∈ (M m (A), id m ⊗ ϕ) which is free with amalgamation over M m (C) with s, is given via its complement in terms of y and the functions…”