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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 $\mathcal{B}=\mathbb{C}$.

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

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