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