The Brown measure of a sum of two free random variables, one of which is triangular elliptic
Serban Belinschi, Zhi Yin, Ping Zhong
Abstract
The triangular elliptic operators are natural extensions of the elliptic deformation of circular operators. We obtain a Brown measure formula for the sum of a triangular elliptic operator $g_{_{α, β, γ}}$ with a random variable $x_0$, which is $*$-free from $g_{_{α, β, γ}}$ with amalgamation over certain unital subalgebra. Let $c_t$ be a circular operator. We prove that the Brown measure of $x_0 + g_{_{α, β, γ}}$ is the push-forward measure of the Brown measure of $x_0 + c_t$ by an explicitly defined map on $\mathbb{C}$ for some suitable $t$. We show that the Brown measure of $x_0+c_t$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{C}$ and its density is bounded by $1/(π{t})$. This work generalizes earlier results on the addition with a circular operator, semicircular operator, or elliptic operator to a larger class of operators. We extend operator-valued subordination functions, due to Biane and Voiculescu, to certain unbounded operators. This allows us to extend our results to unbounded operators.