no. 3
A duality operators/Banach spaces
de la Salle, Mikael1
1Universite Claude Bernard Lyon 1, CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Jean Monnet, I CJ UMR5208, 69622 Villeurbanne, France
Annales de l'Institut Fourier, Volume 76 (2026) no. 3, pp. 1119-1161

To a pair $(T,X)$ of an operator $T$ between subspaces of $L_p$ spaces and a Banach space $X$ we can associate a finite or infinite number, the norm $\Vert T_X \Vert $ of $T$ between the subspaces of the $X$-valued $L_p$ spaces. Given such an operator $T$, we characterize all the operators $S$ for which the implication $\Vert T_X \Vert < \infty \Rightarrow \Vert S_X \Vert <\infty $ holds.

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.

À une paire $(T,X)$ d’un opérateur $T$ entre sous-espaces d’espaces $L_p$ et d’un espace de Banach $X$ on peut associer un nombre, potentiellement infini : la norm $\Vert T_X \Vert $ de $T$ entre sous-espaces d’espaces $L_p$ à valeurs dans $X$. Étant donné un tel opérateur $T$, nous caractérisons tous les autres opérateurs $S$ pour lesquels l’implication $\Vert T_X \Vert < \infty \Rightarrow \Vert S_X \Vert <\infty $ est vraie.

Cet énoncé est une forme du théorème du bipolaire pour une dualité entre classes d’espaces de Banach et classes de sous-espaces d’espaces $L_p$, essentiellement introduite par Gilles Pisier. Nos méthodes permettent également de retrouver l’autre direction – caractériser le bipolaire d’une classe d’espaces de Banach –, qui a été obtenue par Hernandez en 1983.

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Published online:
DOI: 10.5802/aif.3730
Classification: 46A22, 46B20, 46B50
Keywords: duality, geometry of Banach space, bipolar theorem, $L_p$ spaces
Mots-clés : dualité, géometrie des espaces de Banach, théorème du bipolaires, espaces $L_p$

de la Salle, Mikael  1

1 Universite Claude Bernard Lyon 1, CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Jean Monnet, I CJ UMR5208, 69622 Villeurbanne, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
de la Salle, Mikael. A duality operators/Banach spaces. Annales de l'Institut Fourier, Volume 76 (2026) no. 3, pp. 1119-1161. doi: 10.5802/aif.3730
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