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

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 $\ast$-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 $1\mathrm{/}(\pi t)$. 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.