Some progress on the polynomial Dunford–Pettis property

Abstract

A Banach space $E$ has the Dunford-Pettis property (DPP, for short) if every weakly compact (linear) operator on $E$ is completely continuous. In 1979 R. A. Ryan proved that $E$ has the DPP if and only if every weakly compact polynomial on $E$ is completely continuous. Every $k$-homogeneous (continuous) polynomial $P\in \mathcal{P}{(}^{k}E,F)$ between Banach spaces $E$ and $F$ admits an extension $\tilde{P}\in \mathcal{P}{(}^k{E}^{\ast \ast },F^{\ast \ast })$ to the biduals called the Aron–Berner extension. The Aron–Berner extension of every weakly compact polynomial $P\in \mathcal{P}{(}^{k}E,F)$ is $F$-valued, that is, $\tilde{P}(E^{\ast \ast })\subseteq F$, but there are non-weakly compact polynomials with $F$-valued Aron–Berner extension. For Banach spaces $F$ with weak-star sequentially compact dual unit ball $B_{F^{\ast }}$, we strengthen Ryan's result by showing that $E$ has the DPP if and only if every polynomial $P\in \mathcal{P}{(}^{k}E,F)$ with $F$-valued Aron–Berner extension is completely continuous. This gives a partial answer to a question raised in 2003 by I. Villanueva and the second named author. They proved the result for spaces $E$ such that every operator from $E$ into its dual $E^{\ast }$ is weakly compact, but the question remained open for other spaces.