On the Brown measure of $x + i y$, with $x,y$ selfadjoint and $y$ free Poisson

Abstract

Let $x,y$ be freely independent selfadjoint elements in a $W^{*}$-probability space, where $y$ has free Poisson distribution of parameter $p$. We pursue a methodology for computing the Brown measure of $x + i y$, which relies on the matrix-valued subordination function $\Omega$ of the hermitization of $x + i y$, and on the fact that $\Omega$ has an explicitly described left inverse $H$. Our main point is that the Brown measure of $x + i y$ becomes more approachable when it is reparametrized via a certain change of variable $h : \mathcal{D} \to \mathcal{M}$, with $\mathcal{D}, \mathcal{M}$ open subsets of $\mathbb{C}$, where $\\mathcal{D}$ and $h$ are defined in terms of the aforementioned left inverse $H$, and $\mathrm{cl} \,(\mathcal{M})$ contains the support of the absolutely continuous part of Brown measure. More precisely, we find (with some conditions on the distribution of $x$) the following formula: \[ f(s + i \, t) =\frac{1}{2 \pi}\left[\frac{1}{t}\left(\frac{\partial \alpha}{\partial s} +\frac{\partial \beta}{\partial t}\right)-\frac{1}{t}-\frac{\beta}{t^2}\right], \ \ s + i \, t \in \mathcal{M}, \] where $f$ is the density of the absolutely continuous part of the Brown measure and the functions $\alpha, \beta : \mathcal{M} \to \mathbb{R}$ are the real and respectively the imaginary part of $h^{-1}$. We show that if $x$ has an atom $\alpha$ with $\mu_x(\alpha)>p$, then the Brown measure of $x+iy$ has an atom of mass $\mu_x(\alpha)-p$ at the same point $\alpha$. Moreover we prove that if $\alpha_1,\ldots,\alpha_k$ is the list of atoms of $x$ with mass bigger than $p$, then the Brown measure of $x+iy$ is supported on $\mathrm{cl}(\mathcal{M})\cup\{\alpha_1,\ldots,\alpha_k\}$.