The atoms of the free additive convolution of two operator-valued distributions

TL;DR

This paper characterizes the atomic parts of the free additive convolution of two operator-valued distributions and determines eigenvalues of polynomials in freely independent variables using linearization techniques.

Contribution

It provides a detailed description of the atoms in the sum of two free operator-valued distributions and extends eigenvalue analysis to polynomials in these variables.

Findings

Characterization of atoms in free additive convolution

Eigenvalue determination for polynomials in free variables

Application of linearization to operator-valued distributions

Abstract

Suppose that $X\_{1}$ and $X\_{2}$ are two selfadjoint random variables that are freely independent over an operator algebra $\mathcal{B}$. We describe the possible operator atoms of the distribution of $X\_{1}+X\_{2}$ and, using linearization, we determine the possible eigenvalues of an arbitrary polynomial $p(X\_{1},X\_{2})$ in case $\mathcal{B}=\mathbb{C}$.