A duality operators/Banach spaces
Mikael de la Salle

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 of operators between subspaces of spaces, we characterize the operators between subspaces of spaces that remain bounded on the -valued space for every Banach space on which elements of the original class 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 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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Taxonomy
TopicsAdvanced Banach Space Theory · Advanced Harmonic Analysis Research · Approximation Theory and Sequence Spaces
