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The Brown measure of a sum of two free nonselfadjoint random variables, one of which is R-diagonal

Hari Bercovici, Ping Zhong

TL;DR

The paper addresses the Brown measure of the sum $X=X_{1}+X_{2}$ of two $*$-free nonselfadjoint random variables with $X_{2}$ $R$-diagonal. It develops a subordination-based framework that expresses $\mu_{X}$ via additive free convolution data and yields an open set $\Omega$ on which $\mu_{X}$ has a real-analytic density, with the support contained in the closure of $\Omega$. The key tool links $\widetilde{\mu}_{|X|}$ to the symmetrized modulii via $\widetilde{\mu}_{|X_{1}-\lambda|}\boxplus\widetilde{\mu}_{|X_{2}|}$ and uses fixed-point subordination equations for $\omega_{1}^{(\lambda)}(0)$ to compute the logarithmic potential and its Laplacian. The results extend prior work on the circular and $R$-diagonal settings, provide explicit density formulas in several cases, and illustrate the necessity of the full subordination framework through diverse examples, including nilpotent plus Haar unitary and circular perturbations. The findings have implications for nonnormal operator theory and random matrix asymptotics by clarifying when the Brown measure can be explicitly determined from nonselfadjoint components.

Abstract

Suppose that $X_{1}$ and $X_{2}$ are two $*$-free (generally unbounded) random variables with Brown measures $μ_{X_{1}}$ and $μ_{X_{2}}$, respectively. Using properties of classical free additive convolutions, we develop a method for calculating $μ_{X_{1}+X_{2}}$when $X_{2}$ is $R$-diagonal. This method determines a density relative to Lebesgue measure on an open set whose closure contains the support of $μ_{X_{1}+X_{2}}$. Effective calculations are possible in important cases. Biane and Lehner were the first to make significant progress on the problem we consider, even in some cases in which neither $X_{1}$ nor $X_{2}$ is $R$-diagonal. Our examples overlap with theirs, but we emphasize the use of subordination functions. When $X_{2}$ is circular, $μ_{X_{1}+X_{2}}$ was studied earlier using two different approaches, one involving Hamilton-Jacobi equations, and another using standard free probability techniques. Our work extends the second approach.

The Brown measure of a sum of two free nonselfadjoint random variables, one of which is R-diagonal

TL;DR

The paper addresses the Brown measure of the sum of two -free nonselfadjoint random variables with -diagonal. It develops a subordination-based framework that expresses via additive free convolution data and yields an open set on which has a real-analytic density, with the support contained in the closure of . The key tool links to the symmetrized modulii via and uses fixed-point subordination equations for to compute the logarithmic potential and its Laplacian. The results extend prior work on the circular and -diagonal settings, provide explicit density formulas in several cases, and illustrate the necessity of the full subordination framework through diverse examples, including nilpotent plus Haar unitary and circular perturbations. The findings have implications for nonnormal operator theory and random matrix asymptotics by clarifying when the Brown measure can be explicitly determined from nonselfadjoint components.

Abstract

Suppose that and are two -free (generally unbounded) random variables with Brown measures and , respectively. Using properties of classical free additive convolutions, we develop a method for calculating when is -diagonal. This method determines a density relative to Lebesgue measure on an open set whose closure contains the support of . Effective calculations are possible in important cases. Biane and Lehner were the first to make significant progress on the problem we consider, even in some cases in which neither nor is -diagonal. Our examples overlap with theirs, but we emphasize the use of subordination functions. When is circular, was studied earlier using two different approaches, one involving Hamilton-Jacobi equations, and another using standard free probability techniques. Our work extends the second approach.
Paper Structure (4 sections, 16 theorems, 152 equations, 3 figures)

This paper contains 4 sections, 16 theorems, 152 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Logarithmic integrals and subordination
  3. The calculation of $\mu_{X}$
  4. Examples

Key Result

Theorem 2.1

Suppose that $T_{1},T_{2}\in\widetilde{\mathcal{A}}$ are selfadjoint operators free relative to $\tau$, and $T=T_{1}+T_{2}$. If neither $T_{1}$ nor $T_{2}$ is a constant multiple of the identity operator, then there exist unique continuous functions $\omega_{1},\omega_{2}:\overline{\mathbb{C}^{+}}\t and

Figures (3)

  • Figure 3.1: $F=\{0\}$
  • Figure 3.2: $X_{1}$ unitary with spectrum $\{\pm1\}$
  • Figure 3.3: $F$ contains the points $\sqrt{2},\pm i\sqrt{2})$ at distance $1$ from $0$ and $\sqrt{2}$

Theorems & Definitions (41)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 31 more