Generalized Hausdorff and Weighted Mean Matrices as Operators on lp

Two theorems are proved. Theorem 1 establishes sufficient conditions for a generalized Hausdorff matrix H(λ, α) either to be in B(lp) or not to be in B(lp). Theorem 2 shows, inter alia, that if 1 ≤ p < ∞, an > 0, An ≔ a0 + a1 + ••• + an, and An/nan → c > 0, then the weighted mean matrix Ma with weights an is in B(lp) if and only if c < p. There are two examples about cases when the conditions of the theorems are not satisfied. A short proof of the fact that weighted mean matrices are special generalized Hausdorff matrices is also given.