Infinite Toeplitz and Laurent matrices with localized impurities

Abstract

The paper is concerned with the change of the spectra of infinite Toeplitz and Laurent matrices under perturbations in a prescribed finite set of sites. The main result says that the spectrum of a Toeplitz matrix with a non-constant rational symbol is not affected by small localized impurities, while such impurities can nevertheless enlarge the spectrum of the corresponding Laurent matrix. We also study the spectra that may emerge when randomly perturbing Toeplitz or Laurent matrices in a randomly chosen single site.