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In recent years, motivated by computational purposes, the singular value andspectral features of the symmetrization of Toeplitz matrices generated by aLebesgue integrable function have been studied. Indeed, under the assumptionsthat $f$ belongs to $L^1([-\pi,\pi])$ and it has real Fourier coefficients, thespectral and singular value distribution of the matrix-sequence$\{Y_nT_n[f]\}_n$ has been identified, where $n$ is the matrix-size, $Y_n$ isthe anti-identity matrix, and $T_n[f]$ is the Toeplitz matrix generated by $f$.In this note, we consider the multilevel Toeplitz matrix $T_{\bf n}[f]$generated by $f\in L^1([-\pi,\pi]^k)$, $\bf n$ being a multi-index identifyingthe matrix-size, and we prove spectral and singular value distribution resultsfor the matrix-sequence $\{Y_{\bf n}T_{\bf n}[f]\}_{\bf n}$ with $Y_{\bf n}$being the corresponding tensorization of the anti-identity matrix.

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