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

Max Sun Zhou

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.

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

Abstract

We analyze the Brown measure the non-normal operators , where and 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 -transform in the physics literature, to understand the support and the boundary of the Brown measure of . 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 has been explicitly computed. In the general case, we show that the heuristic implies that the boundary of the Brown measure of is an algebraic curve, and provide an algorithm producing a polynomial defining this curve.

Paper Structure

This paper contains 30 sections, 42 theorems, 374 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Brown measure
  4. Brown measure of $X = p + i q$
  5. Quaternions
  6. Quaternionic Green's function
  7. Quaternionic Green's function and Brown measure
  8. Outline for computing Inverse Quaternionic Green's function for $X = p + i q$
  9. (Inverse) Quaternionic Green's function at a complex number
  10. (Inverse) Quaternionic Green's function for a Hermitian operator
  11. (Inverse) Quaternionic Green's function and multiplication by a complex number
  12. Inverse Quaternionic Green's function for $X = p + i q$
  13. Heuristics
  14. Computing $\mathcal{B}_\mathbf{X}$ for $X = p + i q$
  15. Notation and conventions
  16. ...and 15 more sections

Key Result

Theorem 2.4

Let $p, q \in (M, \tau)$ be Hermitian, freely independent, and where $a, b \in (0, 1)$, $\alpha, \alpha', \beta, \beta' \in \mathbb{R}$, $\alpha \neq \alpha'$, and $\beta \neq \beta'$. The Brown measure of $X$ is the convex combination of 4 atoms with another measure $\mu'$: where The Brown measure $\mu$ is supported on $H \cap R$. Further, $\mu'$ has density extending to all $4$ corners of the

Figures (3)

  • Figure 1: ESDs of $X_n = P_n + i Q_n$
  • Figure 2: $X_n = P_n + i Q_n$$\mu_{P_n} \approx (1/3) \delta_{-1} + (1/3) \delta_{0} + (1/3) \delta_{1}$$\mu_{Q_n} = (1 / 2) \delta_0 + (1/2) \delta_1$$n = 10000$
  • Figure 3: $X_n = P_n + i Q_n$$\mu_{P_n} \approx (1/6) \delta_{-1} + (1/3) \delta_{0} + (1/2) \delta_{1}$$\mu_{Q_n} = (3 / 4) \delta_0 + (1/4) \delta_1$$n = 10000$

Theorems & Definitions (96)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 86 more