# Multilevel symmetrized Toeplitz structures and spectral distribution results for the related matrix-sequences

In recent years, motivated by computational purposes, the singular value and spectral features of the symmetrization of Toeplitz matrices generated by a Lebesgue integrable function have been studied. Indeed, under the assumptions that f belongs to L^1([-π,π]) and it has real Fourier coefficients, the spectral and singular value distribution of the matrix-sequence {Y_nT_n[f]}_n has been identified, where n is the matrix-size, Y_n is the anti-identity matrix, and T_n[f] is the Toeplitz matrix generated by f. In this note, we consider the multilevel Toeplitz matrix T_ n[f] generated by f∈ L^1([-π,π]^k), n being a multi-index identifying the matrix-size, and we prove spectral and singular value distribution results for the matrix-sequence {Y_ nT_ n[f]}_ n with Y_ n being the corresponding tensorization of the anti-identity matrix.

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