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 x0, which is *-free from g alpha,beta,gamma with amalgamation over certain unital subalgebra. Let ct be a circular operator. We prove that the Brown measure of x0 + g alpha,beta,gamma is the push -forward measure of the Brown measure of x0 + ct by an explicitly defined map on C for some suitable t. We show that the Brown measure of x0 + ct is absolutely continuous with respect to the Lebesgue measure on 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. (c) 2024 Elsevier Inc. All rights reserved.