Spectrum of the product of Toeplitz matrices with application in probability

TL;DR

This paper investigates the spectrum of products of Toeplitz matrices with continuous, real-valued symbols, establishing convergence results and applying findings to large deviation principles in probability.

Contribution

It proves the convergence of spectra of finite Toeplitz matrix products to the infinite case and explores spectral bounds with an application to Gaussian quadratic forms.

Findings

Spectrum of finite Toeplitz products converges to infinite case spectrum.

An example shows the supremum of the spectrum set may differ from the product of symbols.

Application to large deviation principles for Gaussian quadratic forms.

Abstract

We study the spectrum of the product of two Toeplitz operators. Assume that the symbols of these operators are continuous and real-valued and that one of them is non-negative. We prove that the spectrum of the product of finite section Toeplitz matrices converges to the spectrum of the product of the semi-infinite Toeplitz operators. We give an example showing that the supremum of this set is not always the supremum of the product of the two symbols. Finally, we provide an application in probability which is the first motivation of this study. More precisely, we obtain a large deviation principle for Gaussian quadratic forms.