The Brown measure of a sum of two free nonselfadjoint random variables, one of which is R-diagonal

Abstract

Suppose that $X_{1}$ and $X_{2}$ are two $*$-free (generally unbounded) random variables with Brown measures $\mu_{X_{1}}$ and $\mu_{X_{2}}$, respectively. Using properties of classical free additive convolutions, we develop a method for calculating $\mu_{X_{1}+X_{2}}$when $X_{2}$ is $R$-diagonal. This method determines a density relative to Lebesgue measure on an open set whose closure contains the support of $\mu_{X_{1}+X_{2}}$. Effective calculations are possible in important cases. Biane and Lehner were the first to make significant progress on the problem we consider, even in some cases in which neither $X_{1}$ nor $X_{2}$ is $R$-diagonal. Our examples overlap with theirs, but we emphasize the use of subordination functions. When $X_{2}$ is circular, $\mu_{X_{1}+X_{2}}$ was studied earlier using two different approaches, one involving Hamilton-Jacobi equations, and another using standard free probability techniques. Our work extends the second approach.