The atoms of the free additive convolution of two 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 $\mathcal{B}=\mathbb{C}$.