On certain matrix algebras related to quasi-Toeplitz matrices

Abstract

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 $\{P_0,P_1,P_2,\dots \}$ 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 $[\theta {\alpha }^{n-2},\theta {\alpha }^{n-3},\dots ,\theta ,\alpha ,0,\dots ]$, 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 CQT-Toolbox of Numer.~Algo. 81(2):741--769, 2019.