On certain matrix algebras related to quasi-Toeplitz matrices

Let A_α be the semi-infinite tridiagonal matrix having subdiagonal and superdiagonal unit entries, (A_α)₁₁=α, where α∈ C, and zero elsewhere. A basis ₀,P₁,P₂,…\ of the linear space P_α spanned by the powers of A_α 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 P_α 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 AS as $A=T+K$, where T is Toeplitz and K is compact, it is represented as $A=P+H$, where P∈ P_α 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.