Canonical eigenvalue distribution of multilevel block Toeplitz sequences with non-Hermitian symbols

Let $f:I_k \rightarrow {\mathcal M}_s$ be a bounded symbol with $I_k=(-\pi,\pi)^k$ and ${\mathcal M}_s$ be the linear space of the complex $s\times s$ matrices, $k,s\ge 1$. We consider the sequence of matrices $\{T_n(f)\}$, where $n=(n_1,\ldots,n_k)$, $n_j$ positive integers, $j=1\,\ldots,k$. Let $T_n(f)$ denote the multilevel block Toeplitz matrix of size $\widehat{n} \,s$, $\widehat{n}=\prod_{j=1}^k n_j$, 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 $\{T_n(f)\}$ admits the canonical eigenvalue distribution with the eigenvalue symbol given exactly by $f$ that is $\{T_n(f)\}\sim_\lambda (f, I_k)$. 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 $R(f)$, $R(f)$ being the essential range of $f$. More precisely, if $R(f)$ has empty interior and does not disconnect the complex plane, then $\{T_n(f)\}\sim_\lambda (f, I_k)$. Here we generalize the latter result for the case where the role of $R(f)$ is played by $\bigcup_{j=1}^s R(\lambda_j(f))$, $\lambda_j(f)$, $j=1,\ldots,s$, being the eigenvalues of the matrix-valued symbol $f$. The result is extended to the algebra generated by Toeplitz sequences with bounded symbols. The theoretical findings are confirmed by numerical experiments, which illustrate their practical usefulness.