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.

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

DONATELLI, MARCO;SERRA CAPIZZANO, STEFANO
2012

Abstract

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.
9783034802963
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11383/1736793
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