A duality operators/Banach spaces

TL;DR

This paper characterizes operators between subspaces of Lp spaces that remain bounded on X-valued Lp spaces for all Banach spaces where the original operators are bounded, extending duality concepts in Banach space theory.

Contribution

It introduces a duality framework for operators and Banach spaces, recovering and extending bipolar theorems related to Lp spaces and operator classes.

Findings

Characterization of bounded operators on X-valued Lp spaces

Extension of bipolar theorem to operator and Banach space classes

Recovery of Hernandez's bipolar characterization from 1983

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.