There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random non-hermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a \bubble with wings" in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the non-periodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the in nite-dimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of nite bidiagonal matrices, in nite bidiagonal matrices (\stochastic Toeplitz operators"), nite periodic matrices, and doubly in nite bidiagonal matrices (\stochastic Laurent operators").

This paper is concerned with the change of the spectra of innite Toeplitz and Laurent matrices under perturbations in a prescribed nite set of sites. The main result says that the spectrum of a Toeplitz matrix with a non-constant rational symbol is not aected 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. 1 Introduction and main results Given a complex-valued continuous function a on the complex unit circle T, a 2 C, we consider the matrices T (a) = (a j k ) 1 j;k=1 and L(a) = (a j k ) 1 j;k=1 ; where a k = 1 2 2 Z 0 a(e i ) e ik d; k 2 Z: The matrices T (a) and L(a) are referred to as the Toeplitz matrix and the Laurent matrix with the symbol a, respectively. These matrices induce bounded operators on l 2 (N) and l 2 (Z), respectively. 1 Email: aboet...

This paper provides a new proof of a theorem of Chandler-Wilde, Chonchaiya and Lindner that the spectra of a certain class of infinite, random, tridiagonal matrices contain the unit disc almost surely. It also obtains an analogous result for a more general class of random matrices whose spectra contain a hole around the origin. The presence of the hole forces substantial changes to the analysis.

This report investigates the possible spectra of large, nite dimensional Toeplitz band matrices with perturbations (impurities, uncertainties) in the upper-left mm block. The main result shows that the asymptotic spectrum of such a matrix is not aected by these perturbations, provided they have suciently small norm. This follows from analysis of structured pseudospectra (structured spectral value sets). In contrast, for typical non-Hermitian Toeplitz matrices there exist certain rankone perturbations of arbitrarily small norm that move an eigenvalue away from the asymptotic spectrum in the large-dimensional limit. 1

We describe how to obtain bounds on the spectrum of a non-self-adjoint operator by means of what we call its higher order numerical ranges. We prove some of their basic properties and describe explain how to compute them. We finally use them to obtain new spectral insights into the non-selfadjoint Anderson model in one and two space dimensions.