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 $n$th 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 $\mathcal {QT}_S$, where, instead of representing a generic matrix $A\in\mathcal{QT}_S$ 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 CQT-Toolbox of Numer.~Algo. 81(2):741--769, 2019.