On large Toeplitz band matrices with an uncertain block

Abstract

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