Singular values and eigenvalues of non-Hermitian block Toeplitz matrices

The asymptotic distribution of singular values and eigenvalues of non-Hermitian block Toeplitz matrices is studied. These matrices are associated with the Fourier series of an univariate function f. The asymptotic distribution of singular values is computed when f belongs to L2 and is matrix-valued, not necessarily square. Clusters of singular values are also studied, and a new result is proved. Moreover, a classical formula due to Szegö concerning the asymptotic spectrum of Hermitian Toeplitz matrices is extended to the non-Hermitian block case, under the assumption that f is bounded and test functions are harmonic. Finally, it is proved that the class of harmonic test functions is optimal, as far as that formula is concerned.