Some progress on the polynomial Dunford-Pettis property.

A Banach space E has the Dunford-Pettis property (DPP, for short) if every weakly compact (linear) operator on E is completely continuous. The L1 and the L∞-spaces have the DPP. 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∈P(kE, F) between Banach spaces E and F admits an extension ̃P∈ P(kE∗∗, F∗∗) called the Aron-Berner extension. The Aron-Berner extension of every weakly compact polynomial P∈P(kE, F) is F-valued, that is, ̃P(E∗∗)⊆F, but there are non weakly compact polynomials with F-valued Aron-Berner extension.We strengthen Ryan’s result by showing that E has the DPP if and only if every polynomial P∈ P(kE, F) with F-valued Aron-Berner extension is completely continuous. This answers a question raised in 2003 by I. Villanueva and the second named author. They proved the result for certain spaces E, for instance, the L∞-spaces, but the question remained open for other spaces such as the L1-spaces.