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Singular values and eigenvalues of non-hermitian block Toeplitz matrices

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Abstract

We study the asymptotic distribution of singular values and eigenvalues of non-Hermitian block Toeplitz matrices, generated by a matrix-valued periodic functionf, supposed to beL 2. A distribution result concerning singular values, due to Avram, Parter and Tyrtyshnikov, is extended to the case wheref is matrix-valued, not necessarily square. Although Szegö's formula no longer holds iff is not Hermitian, we show that the same distribution formula still applies, virtually unchanged, provided test functions are supposed to be harmonic and not only continuous. Moreover, the class of harmonic test functions is proved to be optimal, as far as that formula is concerned.

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  1. Scuola Normale Superiore, Piazza Cavalieri 7, 56100, Pisa, Italy

    Paolo Tilli

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  1. Paolo Tilli
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Tilli, P. Singular values and eigenvalues of non-hermitian block Toeplitz matrices. Calcolo 33, 59–69 (1996). https://doi.org/10.1007/BF02575707

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