On certain matrix algebras related to quasi-Toeplitz matrices

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