Spectral properties of random perturbations of non-normal Toeplitz matrices

Abstract

Spectral properties of Toeplitz operators and their finite truncations have long been central in operator theory. In the finite dimensional, non-normal setting, the spectrum is notoriously unstable under perturbations. Random perturbations provide a natural framework for studying this instability and identifying spectral features that emerge in typical noisy situations. This article surveys recent advances on the spectral behavior of (polynomially vanishing) random perturbations of Toeplitz matrices, focusing mostly on the limiting spectral distribution, the distribution of outliers, and localization of eigenvectors, and highlight the major techniques used to study these problems. We complement the survey with new results on the limiting spectral distribution of polynomially vanishing random perturbation of Toeplitz matrices with continuous symbols, on the limiting spectral distribution of finitely banded Toeplitz matrices under exponentially and super-exponentially vanishing random perturbations, and on the complete localization of outlier eigenvectors of randomly perturbed Jordan blocks.