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 absolutely continuous part of 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 Brown measure. More precisely, we find (with some conditions on the distribution of $x$, which have to be imposed for certain values of the parameter $p$) the following formula: \[ f(s + i \, t) =\frac{1}{4\pi}\left[\frac{2}{t}\left(\frac{\partial \alpha}{\partial s} +\frac{\partial \beta}{\partial t}\right)-\frac{2}{t}-\frac{2\beta}{t^2}\right], \ \ s + i \, t \in \mathcal{M}, \] 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 $h^{-1}$.
| Comments: | Added: a subsection about homeomorphism, some references and comments. Minor corrections |
| Subjects: | Operator Algebras (math.OA); Mathematical Physics (math-ph); Probability (math.PR) |
| Cite as: | arXiv:2512.23528 [math.OA] |
| (or arXiv:2512.23528v2 [math.OA] for this version) | |
| https://doi.org/10.48550/arXiv.2512.23528 arXiv-issued DOI via DataCite |
From: Kamil Szpojankowski [view email]
[v1]
Mon, 29 Dec 2025 15:06:59 UTC (465 KB)
[v2]
Tue, 26 May 2026 14:34:36 UTC (468 KB)
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