Quaternionic Green's Function and the Brown Measure of Atomic Operators

We analyze the Brown measure the non-normal operators $X = p + i q$, where $p$ and $q$ are Hermitian, freely independent, and have spectra consisting of finitely many atoms. We use the Quaternionic Green's function, an analogue of the operator-valued $R$-transform in the physics literature, to understand the support and the boundary of the Brown measure of $X$. We present heuristics for the boundary and support of the Brown measure in terms of the Quaternionic Green's function and verify they are true in the cases when the Brown measure of $X$ has been explicitly computed. In the general case, we show that the heuristic implies that the boundary of the Brown measure of $X$ is an algebraic curve, and provide an algorithm producing a polynomial defining this curve.