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

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.

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