A note on the analytic complete continuity property

We show that a Banach space X has the analytic complete continuity property if and only if for every 1 less than or equal to p < infinity and for every f is an element of H-p(D; X), the sequence f(r(n)e(l)) is p-Pettis-Cauchy for every r(n) up arrow 1. This allows us to show that X has the analytic complete continuity property if and only if L-p(Omega;X) has this property for every 1 less than or equal to p < infinity and for every sigma-finite measure space (Omega, Sigma, mu). (C) 2002 Elsevier Science.