On certain matrix algebras related to quasi-Toeplitz matrices

Let Aα be the semi-infinite tridiagonal matrix having subdiagonal and superdiagonal unit entries, (Aα)11=α, where α∈C, and zero elsewhere. A basis {P0,P1,P2,…} of the linear space Pα spanned by the powers of Aα is determined, where P0=I, Pn=Tn+Hn, Tn is the symmetric Toeplitz matrix having ones in the nth super- and sub-diagonal, zeros elsewhere, and Hn is the Hankel matrix with first row [θαn-2,θαn-3,…,θ,α,0,…], where θ=α2-1. The set Pα is an algebra, and for α∈{-1,0,1}, Hn has only one nonzero anti-diagonal. This fact is exploited to provide a better representation of symmetric quasi-Toeplitz matrices QTS, where, instead of representing a generic matrix A∈QTS as A=T+K, where T is Toeplitz and K is compact, it is represented as A=P+H, where P∈Pα 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.