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

TL;DR

This paper introduces the Quaternionic Green's function as a tool to analyze the Brown measure of atomic operators, revealing that the boundary of the measure is an algebraic curve and providing an algorithm to determine this boundary.

Contribution

It develops a novel approach using Quaternionic Green's function to understand the Brown measure of non-normal operators with atomic spectra, including an algorithm for boundary determination.

Findings

The boundary of the Brown measure is an algebraic curve.

Heuristics for support and boundary are validated in explicit cases.

An algorithm for computing the polynomial defining the boundary curve is provided.

Abstract

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.