A note on summability in Banach spaces

Let Z and X be Banach spaces. Suppose that X is Asplund. Let M be a bounded set of operators from Z to X with the following property: a bounded sequence (zₙ)n∈ N in Z is weakly null if, for each M ∈ M, the sequence (M(zₙ))n∈ N is weakly null. Let (zₙ)n∈ N be a sequence in Z such that: (a) for each n∈ N, the set (zₙ):M∈ M\ is relatively norm compact; (b) for each sequence (Mₙ)n∈ N in M, the series ∑ₙ₌₁^∞ Mₙ(zₙ) is weakly unconditionally Cauchy. We prove that if T∈ M is Dunford-Pettis and infn∈ N\|T(zₙ)\|\|zₙ\|⁻¹>0, then the series ∑ₙ₌₁^∞ T(zₙ) 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.