The Brown Measure of Non-Hermitian Sums of Projections

TL;DR

This paper calculates the Brown measure for a class of non-normal operators formed by sums of free projections, revealing support on hyperbolas and properties related to atoms and symmetries.

Contribution

It introduces a method to compute the Brown measure of sums of free projections with two-atom spectra, linking free probability with random matrix models.

Findings

Brown measures supported on hyperbolas

Identification of spectral properties related to atoms

Symmetry properties of the measures

Abstract

We compute the Brown measure of the non-normal operators $X = p + i q$, where $p$ and $q$ are Hermitian, freely independent, and have spectra consisting of $2$ atoms. The computation relies on the model of the non-trivial part of the von Neumann algebra generated by 2 projections as $2 \times 2$ random matrices. We observe that these measures are supported on hyperbolas and note some other properties related to their atoms and symmetries.