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

Second version, major modifications. Comments most welcome