Toeplitz and Hankel matrices on \(\mathbb{C}^n\) (Q5945063)

The author suggests some characterizations of Toeplitz and Hankel matrices on \(\mathbb{C}^n\). Let \(A_n\) be an (\(n\times n\))-matrix in \(\mathcal{B}(\mathbb{C}^n)\) and \(V_n\) the (\(n\times n)\)-matrix with the entries \(a_{ij}=1\) when \ \(j-i=1, \;i=1,2,\dots ,n-1,\) and \(a_{ij}=0\) otherwise. The following statements, for example, are proved: 1. \(A_n\) is a Toeplitz matrix if and only if \(V_nV_n^\ast A_n V_nV_n^\ast = V_nA_nV_n^\ast\); 2. if \(A_n\) commutes with \(V_n+e^{i\theta}V_n^{\ast n-1}\) for any \(\theta\in [0,2\pi)\), then \(A_n\) is a Toeplitz matrix. Some characterization of Toeplitz matrices in terms of the algebra \(\mathcal{R}(V_n+e^{i\theta}V_n^{\ast n-1})\) is also given. Similar statements are also proved for Hankel matrices.