On certain matrix algebras related to quasi-Toeplitz matrices

Let \(A_\alpha \) be the semi-infinite tridiagonal matrix having subdiagonal and superdiagonal unit entries, \((A_\alpha )_{11}=\alpha \), where \(\alpha \in \mathbb C\), and zero elsewhere. A basis \(\{P_0,P_1,P_2,\ldots \}\) of the linear space \(\mathcal {P}_\alpha \) spanned by the powers of \(A_\alpha \) is determined, where \(P_0=I\), \(P_n=T_n+H_n\), \(T_n\) is the symmetric Toeplitz matrix having ones in the nth super- and sub-diagonal, zeros elsewhere, and \(H_n\) is the Hankel matrix with first row \([\theta \alpha ^{n-2}, \theta \alpha ^{n-3}, \ldots , \theta , \alpha , 0, \ldots ]\), where \(\theta =\alpha ^2-1\). The set \(\mathcal {P}_\alpha \) is an algebra, and for \(\alpha \in \{-1,0,1\}\), \(H_n\) has only one nonzero anti-diagonal. This fact is exploited to provide a better representation of symmetric quasi-Toeplitz matrices \(\mathcal{Q}\mathcal{T}_S\), where, instead of representing a generic matrix \(A\in \mathcal{Q}\mathcal{T}_S\) as \(A=T+K\), where T is Toeplitz and K is compact, it is represented as \(A=P+H\), where \(P\in \mathcal {P}_\alpha \) and H is compact. It is shown experimentally that the matrix arithmetic obtained this way is much more effective than that implemented in the toolbox CQT-Toolbox of Numer. Algo. 81(2):741–769, 2019.