Optimal domain of $q$-concave operators and vector measure representation of $q$-concave Banach lattices

TL;DR

This paper characterizes the maximal domain extension of $q$-concave operators on quasi-Banach function spaces and provides a representation theorem for $q$-concave Banach lattices via vector measures.

Contribution

It introduces the concept of the optimal domain for $q$-concave operators and establishes a new representation theorem for $q$-concave Banach lattices using vector measures.

Findings

Existence of maximal $q$-concave extensions for operators.

A new representation theorem for $q$-concave Banach lattices.

Connection between $q$-concave operators and vector measure spaces.

Abstract

Given a Banach space valued $q$-concave linear operator $T$ defined on a $\sigma$-order continuous quasi-Banach function space, we provide a description of the optimal domain of $T$ preserving $q$-concavity, that is, the largest $\sigma$-order continuous quasi-Banach function space to which $T$ can be extended as a $q$-concave operator. We show in this way the existence of maximal extensions for $q$-concave operators. As an application, we show a representation theorem for $q$-concave Banach lattices through spaces of integrable functions with respect to a vector measure. This result culminates a series of representation theorems for Banach lattices using vector measures that have been obtained in the last twenty years.