On certain matrix algebras related to quasi-Toeplitz matrices

Let A_α A α be the semi-infinite tridiagonal matrix having subdiagonal and superdiagonal unit entries, (A_α )₁₁=α ( A α ) 11 = α , where α ∈ C α ∈ C , and zero elsewhere. A basis ₀,P₁,P₂,… \ { P 0 , P 1 , P 2 , … } of the linear space P_α P α spanned by the powers of A_α A α is determined, where P₀=I P 0 = I , Pₙ=Tₙ+Hₙ P n = T n + H n , Tₙ T n is the symmetric Toeplitz matrix having ones in the n th super- and sub-diagonal, zeros elsewhere, and Hₙ H n is the Hankel matrix with first row [θ α ⁿ⁻², θ α ⁿ⁻³, … , θ , α , 0, … ] [ θ α n - 2 , θ α n - 3 , … , θ , α , 0 , … ] , where θ =α ²-1 θ = α 2 - 1 . The set P_α P α is an algebra, and for α ∈ \-1,0,1\ α ∈ { - 1 , 0 , 1 } , Hₙ H n has only one nonzero anti-diagonal. This fact is exploited to provide a better representation of symmetric quasi-Toeplitz matrices QTS Q T S , where, instead of representing a generic matrix A∈ QTS A ∈ Q T S as $$A=T+K$$ A = T + K , where T is Toeplitz and K is compact, it is represented as $$A=P+H$$ A = P + H , where P∈ P_α 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 toolbox of Numer. Algo. 81(2):741–769, 2019.