A duality operators/Banach spaces
[Une dualité entre opérateurs et espaces de Banach]
de la Salle, Mikael1
1 Universite 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, Online first, 43 p.

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 ∥T X ∥ 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 ∥T X ∥<∞⇒∥S X ∥<∞ 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 ∥T X ∥ 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 ∥T X ∥<∞⇒∥S X ∥<∞ 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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Révisé le :
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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 
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de la Salle, Mikael. A duality operators/Banach spaces. Annales de l'Institut Fourier, Online first, 43 p.

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