A duality operators/Banach spaces

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.