A Note on the Analytic Complete Continuity Property1

We show that a Banach space X has the analytic complete continuity property if and only if for every 1 ≤ p < ∞ and for every f ∈ Hp X , the sequence f rnei· is p-Pettis–Cauchy for every rn ↑ 1.This allows us to show that X has the analytic complete continuity property if and only if Lp X has this property for every 1 ≤ p < ∞ and for every σ-finite measure space ( μ).