The eigenvalue distribution of products of Toeplitz matrices - clustering and attraction
Based on an earlier result of \textit{L. Golinskii} and \textit{S. Serra-Capizzano} [J. Approximation Theory, 144, No. 1, 84--102 (2007; Zbl 1111.15013)] establishing a link between distributions of non-Hermitian perturbations of Hermitian matrix sequences and the distribution of the original sequence, in this paper it is proved that the asymptotic spectrum of the product of Toeplitz sequences, whose symbols have a real-valued essentially bounded product \(h\), is described by the function \(h\) in the ''Szegö way''. Further, using Mergelyan's theorem, this result is extended to the more general case where \(h\) belongs to the Tilli class. An analogous result is obtained for sequences belonging to the algebra generated by Toeplitz sequences, if the symbols associated with the sequences are bounded and the global symbol \(h\) belongs to the Tilli class. Finaly, a generalization to the case of multilevel matrix-valued symbols is derived and the case of Laurent polynomials not necessarily belonging to the Tilli class is studied.
scientific article; zbMATH DE number 5689430
scientific article