JournalsrlmVol. 29, No. 4pp. 679–687

Some progress on the polynomial Dunford–Pettis property

  • Raffaella Cilia

    Università degli Studi di Catania, Italy
  • Joaquín M. Gutiérrez

    Universidad Politécnica de Madrid, Spain
Some progress on the polynomial Dunford–Pettis property cover
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Abstract

A Banach space EE has the Dunford-Pettis property (DPP, for short) if every weakly compact (linear) operator on EE is completely continuous. In 1979 R. A. Ryan proved that EE has the DPP if and only if every weakly compact polynomial on EE is completely continuous. Every kk-homogeneous (continuous) polynomial PP(k ⁣E,F)P\in\mathcal P(^k\!E,F) between Banach spaces EE and FF admits an extension P~P(k ⁣E,F)\widetilde P\in\mathcal P(^k\!E^{**},F^{**}) to the biduals called the Aron–Berner extension. The Aron–Berner extension of every weakly compact polynomial PP(k ⁣E,F)P\in{\mathcal P}(^k\!E,F) is FF-valued, that is, P~(E)F\widetilde P(E^{**})\subseteq F, but there are non-weakly compact polynomials with FF-valued Aron–Berner extension. For Banach spaces FF with weak-star sequentially compact dual unit ball BFB_{F^*}, we strengthen Ryan's result by showing that EE has the DPP if and only if every polynomial PP(k ⁣E,F)P\in\mathcal P(^k\!E,F) with FF-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 EE such that every operator from EE into its dual EE^* is weakly compact, but the question remained open for other spaces.

Cite this article

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. 679–687

DOI 10.4171/RLM/828