The Brown measure of a sum of two free random variables, one of which is triangular elliptic

TL;DR

This paper derives a formula for the Brown measure of the sum of a triangular elliptic operator and a free random variable, extending previous results to a broader class of operators and unbounded cases.

Contribution

It generalizes the Brown measure calculation for sums involving triangular elliptic operators and extends subordination techniques to unbounded operators.

Findings

Brown measure of the sum is a push-forward of the measure with a circular operator

The measure is absolutely continuous with a bounded density

Results extend to unbounded operators using operator-valued subordination

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 $g_{_{\alpha, \beta, \gamma}}$ with a random variable $x_0$, which is $*$-free from $g_{_{\alpha, \beta, \gamma}}$ with amalgamation over certain unital subalgebra. Let $c_t$ be a circular operator. We prove that the Brown measure of $x_0 + g_{_{\alpha, \beta, \gamma}}$ 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 $1/(\pi{t})$. 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.