# Journal of Operator Theory

Volume 85, Issue 1, Winter 2021 pp. 303-320.

The atoms of operator-valued free convolutions

**Authors**:
Serban T. Belinschi (1), Hari Bercovici (2), Weihua Liu (3)

**Author institution:** (1) Institute de Mathematiques de Toulouse; UMR5219;
Universite de Toulouse; CNRS; UPS, F-31062 Toulouse, France

(2) Department of Mathematics,
Indiana University,
Bloomington, IN 47405, U.S.A.

(3) Department of Mathematics,
The University of Arizona,
617 N. Santa Rita Ave.,
P.O. Box 210089,
Tucson, AZ 85721-0089 U

**Summary: **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 $\mathcal{B}=\mathbb{C}$.

**DOI: **http://dx.doi.org/10.7900/jot.2019dec07.2283

**Keywords: **free probability, operator-valued distributions, applications of linearization/realization of polynomials

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