Cesaro-Like Operators - Rhoades, B. E.; Trutt, D. (Sarajevo Journal of Mathematics, 2020)
In previous work it was shown that the lower triangular generalized Hausdorff matrix H_α with nonzero entries h_nk = (n+α+1)^-1, for α ≥ 0, is subnormal on ^2 if and only if α = 0, 1, 2, …. For 0 < h ≤ 1, the weighted Cesaro operator C^′_h : {a_n} → {b_n} on ^2, when b_n = (a_0 + d_1a_1 + … + d_na_n)/(n + 1)d_n, is subnormal when d_j^2 = Γ(j + 1)Γ(h)/Γ(j + h). In this paper we show that, when d_j = Γ(j + 1)Γ(h)/Γ(j + h), the square of the weights chosen above, then the corresponding operator C_h is bounded on ^2 for 0 < h < 3/2, that H_α is bounded on ^2 for all non-integer α < 0, and that C_h is closely related to H_h-1. This relationship leads to our main result that C_h is only subnormal when h = 1, when it corresponds to the original Cesaro operator with α = 0 and each d_j = 1.