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The spectra of polynomials in free (semi)circular operators

Akihiro Miyagawa

Abstract

We show that any $L^2$-bounded rational function in free semicircular random variables is a bounded operator, which implies the coincidence of the usual spectrum and $L^2$-spectrum for rational functions. Based on this observation, we also compute the spectra of several polynomials in free circular random variables.

The spectra of polynomials in free (semi)circular operators

Abstract

We show that any -bounded rational function in free semicircular random variables is a bounded operator, which implies the coincidence of the usual spectrum and -spectrum for rational functions. Based on this observation, we also compute the spectra of several polynomials in free circular random variables.
Paper Structure (4 sections, 14 theorems, 104 equations, 1 figure)

This paper contains 4 sections, 14 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Main theorem
  4. Computation of the spectrum for free circular elements

Key Result

Theorem 1.1

If $f \in D(s) \cap L^2(s,\tau)$, then $f \in L^{\infty}(s,\tau)$.

Figures (1)

  • Figure 1: For each polynomial $P$, the area of $\lambda \in \mathbb{C}$ where the spectral radius $r(Q_{\lambda})<1$ is colored in gray, and the red line describes its boundary. The eigenvalues of corresponding Ginibre models of size $N=3000$ are plotted in blue.

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Theorem 1.3: Theorem 4.4 in MR1784419
  • Theorem 1.4
  • Remark 1.5
  • Theorem 2.1: Theorem 3.1 and 4.5 in MR4452070
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 20 more