A note on summability in Banach spaces

Abstract

Let $Z$ and $X$ be Banach spaces. Suppose that $X$ is Asplund. Let $\mathcal{M}$ be a bounded set of operators from $Z$ to $X$ with the following property: a bounded sequence $(z_n{)}_{n\in \mathbb{N}}$ in $Z$ is weakly null if, for each $M\in \mathcal{M}$, the sequence $(M(z_n){)}_{n\in \mathbb{N}}$ is weakly null. Let $(z_n{)}_{n\in \mathbb{N}}$ be a sequence in $Z$ such that: (a) for each $n\in \mathbb{N}$, the set $\{M(z_n):M\in \mathcal{M}\}$ is relatively norm compact; (b) for each sequence $(M_n{)}_{n\in \mathbb{N}}$ in $\mathcal{M}$, the series ${\sum }_{n=1}^{\infty }{M}_n(z_n)$ is weakly unconditionally Cauchy. We prove that if $T\in \mathcal{M}$ is Dunford-Pettis and ${\inf }_{n\in \mathbb{N}}\Vert T(z_n)\Vert \Vert z_n{\Vert }^{-1}>0$, then the series ${\sum }_{n=1}^{\infty }T(z_n)$ is absolutely convergent. As an application, we provide another proof of the fact that a countably additive vector measure taking values in an Asplund Banach space has finite variation whenever its integration operator is Dunford-Pettis.