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.

翻译：设 $A_\alpha$ 为半无穷三对角矩阵，其次对角线与超对角线元素均为单位1，$(A_\alpha)_{11}=\alpha$（$\alpha\in\mathbb C$），其余元素为零。在由 $A_\alpha$ 的幂张成的线性空间 $\mathcal P_\alpha$ 中，确定了基 $\{P_0,P_1,P_2,\ldots\}$，其中 $P_0=I$，$P_n=T_n+H_n$，$T_n$ 为在第 $n$ 条超对角线与次对角线上元素为1、其余为零的对称Toeplitz矩阵，$H_n$ 为第一行为 $[\theta\alpha^{n-2}, \theta\alpha^{n-3}, \ldots, \theta, \alpha, 0, \ldots]$（$\theta=\alpha^2-1$）的Hankel矩阵。集合 $\mathcal P_\alpha$ 构成代数结构，且当 $\alpha\in\{-1,0,1\}$ 时，$H_n$ 仅含一条非零反对角线。利用此性质可更优地表征对称拟Toeplitz矩阵 $\mathcal {QT}_S$：不再将一般矩阵 $A\in\mathcal{QT}_S$ 表示为 $A=T+K$（其中 $T$ 为Toeplitz矩阵，$K$ 为紧算子），而是表示为 $A=P+H$（其中 $P\in\mathcal P_\alpha$，$H$ 为紧算子）。实验表明，由此获得的矩阵算术效率远高于 Numer.~Algo. 81(2):741--769, 2019 中CQT-Toolbox的实现。