Abstract
Let f : Ik→Ms 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.
Mathematics Subject Classification (2000). Primary 47B35, Secondary 15A18.
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Donatelli, M., Neytcheva, M., Serra-Capizzano, S. (2012). Canonical Eigenvalue Distribution of Multilevel Block Toeplitz Sequences with Non-Hermitian Symbols. In: Arendt, W., Ball, J., Behrndt, J., Förster, KH., Mehrmann, V., Trunk, C. (eds) Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations. Operator Theory: Advances and Applications, vol 221. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0297-0_15
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