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Canonical Eigenvalue Distribution of Multilevel Block Toeplitz Sequences with Non-Hermitian Symbols

  • Marco DonatelliEmail author
  • Maya Neytcheva
  • Stefano Serra-Capizzano
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)

Abstract

Let f : IkMs be a bounded symbol with Ik = (-𝜈, 𝜋)k andMs be the linear space of the complex s × s matrices, k, s ≥ 1.W e consider the sequence of matrices {Tn( )}, where n = (n1,... , nk), nj positive integers, j = 1 ... , k.Let Tn(f) denote the multilevel block Toeplitz matrix of size ňs, ň = ∏K j=1 nj, constructed in the standard way by using the Fourier coefficients of the symbol f.If f is Hermitian almost everywhere, then it is well known that {Tn(f)} admits the canonical eigenvalue distribution with the eigenvalue symbol given exactly by f that is {Tn(f)} ~ λ (f, Ik).When s = 1, thanks to the work of Tilli, more about the spectrum is known, independently of the regularity of f and relying only on the topological features of (f), R(f) being the essential range of. More precisely, if Rf(Rf) has empty interior and does not disconnect the complex plane, then {T n(f)} ∼ λ (f, T k).Here we generalize the latter result for the case where the role of (f) is played being the eigenvalues of the matrixvalued symbol f.Th e result is extended to the algebra generated by Toeplitz sequences with bounded symbols.T he theoretical findings are confirmed by numerical experiments, which illustrate their practical usefulness.

Keywords

Matrix sequence joint eigenvalue distribution Toeplitz matrix 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  • Marco Donatelli
    • 1
    Email author
  • Maya Neytcheva
    • 2
  • Stefano Serra-Capizzano
    • 1
  1. 1.Dipartimento di Scienza ed alta TecnologiaUniversità dell InsubriaComoItaly
  2. 2.Department of Information TechnologyUppsala UniversityUppsalaSweden

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