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A Banach space has the Dunford-Pettis property (DPP, for short) if every weakly compact (linear) operator on is completely continuous. In 1979 R. A. Ryan proved that has the DPP if and only if every weakly compact polynomial on is completely continuous. Every -homogeneous (continuous) polynomial between Banach spaces and admits an extension to the biduals called the Aron–Berner extension. The Aron–Berner extension of every weakly compact polynomial is -valued, that is, , but there are non-weakly compact polynomials with -valued Aron–Berner extension. For Banach spaces with weak-star sequentially compact dual unit ball , we strengthen Ryan's result by showing that has the DPP if and only if every polynomial with -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 such that every operator from into its dual is weakly compact, but the question remained open for other spaces.
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Raffaella Cilia, Joaquín M. Gutiérrez, Some progress on the polynomial Dunford–Pettis property. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), no. 4, pp. 679–687DOI 10.4171/RLM/828