A duality operators/Banach spaces

2021/01/19 by Mikael de la Salle, de la Salle, Mikael
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2101.07666

openalex publication_date 2021/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a set B of operators between subspaces of L_p spaces, we characterize the operators between subspaces of L_p spaces that remain bounded on the X-valued L_p space for every Banach space on which elements of the original class B are bounded. This is a form of the bipolar theorem for a duality between the class of Banach spaces and the class of operators between subspaces of L_p spaces, essentially introduced by Pisier. The methods we introduce allow us to recover also the other direction --characterizing the bipolar of a set of Banach spaces--, which had been obtained by Hernandez in 1983.

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