The spectra of polynomials in free (semi)circular operators

TL;DR

This paper proves that rational functions in free semicircular variables are bounded operators and computes spectra of polynomials in free circular variables, linking $L^2$-spectra with usual spectra.

Contribution

It establishes the boundedness of $L^2$-bounded rational functions in free semicircular variables and computes spectra for polynomials in free circular variables.

Findings

$L^2$-bounded rational functions are bounded operators

Spectra of rational functions coincide with $L^2$-spectra

Spectra of certain polynomials in free circular variables are explicitly computed

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.