A CHARACTERIZATION OF ABSOLUTE SUMMABILITY FACTORS

Abstract

Let $A$ and $B$ be two summability methods. We shall use the notation $\lambda \in(A, B)$ to denote the set of all sequences $\lambda$ such that $\sum\nolimits a_{n}\lambda_{n}$ is summable $B$, whenever $\sum\nolimits a_{n}$ is summable $A$. In the present paper we characterize the sets $\lambda \in (|\overline{N}, p_{n}|, |T|_{k})$ and $\lambda \in (|\overline{N}, p_{n}|_{k}, |T|)$, where $T$ is a lower triangular matrix with positive entries and row sums $1$. As special cases we obtain summability factor theorems and inclusion theorems for pairs of weighted mean matrices.