The eigenvalue distribution of products of Toeplitz matrices - Clustering and attraction |
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Authors: | Stefano Serra-Capizzano Debora Sesana |
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Institution: | a Department of Physics and Mathematics, University of “Insubria”, Via Valleggio 11, 22100 Como, Italy b Department of Mathematics, University of Bordeaux I, Cours de la Liberation, 33405 Talence-Cedex, France |
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Abstract: | We use a recent result concerning the eigenvalues of a generic (non-Hermitian) complex perturbation of a bounded Hermitian sequence of matrices to prove 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”. Then, using Mergelyan’s theorem, we extend the result to the more general case where h belongs to the Tilli class. The same technique gives us the analogous result 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. A generalization to the case of multilevel matrix-valued symbols and a study of the case of Laurent polynomials not necessarily belonging to the Tilli class are also given. |
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Keywords: | 15A18 15A12 47B36 47B65 |
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