Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra

We obtain, under mild assumptions, a formula for the Brown measure of the sum of a twisted elliptic operator $g_{t,\gamma}$ with an arbitrary random variable $x_0$, freely independent from $g_{t,\gamma}$. The twisted elliptic operators include Voiculescu's circular operator and elliptic operators. We show that the Brown measure of the sum of a twisted elliptic operator with a free random variable is the push-forward measure of the Brown measures of the sum of a free circular with the same free random variable under a natural map, provided that this map is one-to-one and non-singular. This generalizes earlier results about free additive Brownian motions where the free random variable $x_0$ is assumed to be selfadjoint. In particular, we prove that the Brown measure of the sum of a Haar unitary operator and a twisted elliptic element is supported in a deformed ring where the inner boundary is a circle and the outer boundary is an ellipse, and its density is constant along a family of ellipses. The approach is based on a Hermitian reduction and subordination functions.