| 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 gα,β,γ with a random variable x0, which is ⁎-free from gα,β,γ with amalgamation over certain unital subalgebra. Let ct be a circular operator. We prove that the Brown measure of x0+gα,β,γ 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/(πt). 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. |
, Ping Zhong 
