The spectra of polynomials in free (semi)circular operators

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.

arXiv comment23 pages and 4 figures. This is the 2nd version. We fixed typos and added Corollary 4.2