The spectra of polynomials in free (semi)circular operators
Operator Algebras
2026-04-22 v2 Functional Analysis
Probability
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.
Keywords
Cite
@article{arxiv.2603.19528,
title = {The spectra of polynomials in free (semi)circular operators},
author = {Akihiro Miyagawa},
journal= {arXiv preprint arXiv:2603.19528},
year = {2026}
}
Comments
23 pages and 4 figures. This is the 2nd version. We fixed typos and added Corollary 4.2