Extreme singular values and eigenvalues of non-Hermitian block Toeplitz matrices

In this paper we are concerned with the analysis of the distribution and localization of the singular values of Toeplitz matrices {T-n(f)} generated by a p-variate Lebesgue integrable matrix-valued function f:Q(p) --> C-hxk, Q = (- pi,pi). We prove that the union of the essential ranges of the singular values of f is a proper/weak cluster for the whole set of the singular values of {T-n(f)}, by showing that the number of outliers is strongly depending on the regularity features of the underlying function f: in particular, if f is continuous or fi-om the Krein algebra and p = 1, then the cluster is proper. Other results concerning the extreme spectral behavior of {T-n(f)}, second-order ergodic formulas and localization of eigenvalues of preconditioned matrices {Tn(-1)(g)T-n(f)} are presented. Some examples of applications to the preconditioning of these results are also discussed. (C) 1999 Elsevier Science B.V. All rights reserved.