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 $\ast$-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.

25 pages, 4 figures, completely rewritten