On kth-Order Slant Weighted Toeplitz Operator

Let β = {β (n)}(n∈ℤ) be a sequence of positive numbers with β (0) = 1, 0 < β (n)/β (n+1) ≤ 1 when n ≥ 0 and 0 < β (n)/β (n−1) ≤ 1 when n ≤ 0. A kth-order slant weighted Toeplitz operator on L (2)(β) is given by U (ϕ) = W (k) M (ϕ), where M (ϕ) is the multiplication on L (2)(β) and W (k) is an operator on L (2)(β) given by W (k) e (nk)(z) = (β (n)/β (nk))e (n)(z), {e (n)(z) = z (k)/β (k)}(k∈ℤ) being the orthonormal basis for L (2)(β). In this paper, we define a kth-order slant weighted Toeplitz matrix and characterise U (ϕ) in terms of this matrix. We further prove some properties of U (ϕ) using this characterisation.