Banach Lattice Structures and Concavifications in Banach Spaces

L Agud ,M A Juan ,J M Calabuig ,Enrique Alfonso Sánchez Pérez

doi:10.3390/math8010127

Abstract

Let ( Ω , Σ , μ ) be a finite measure space and consider a Banach function space Y ( μ ) . We say that a Banach space E is representable by Y ( μ ) if there is a continuous bijection I : Y ( μ ) → E . In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces.

Highlights

The theory of Banach lattices, and in particular the theory of Banach function spaces, provides powerful specific tools in mathematical analysis
The main notion that we develop is the pth power of a Banach space
This provides a first representation result, using a linear isomorphism in this case: A Banach space is representable as an order-continuous Banach function space with a weak unit if and only if it is the range of an integration map of a vector measure which is an isomorphism

Summary

Introduction

The theory of Banach lattices, and in particular the theory of Banach function spaces, provides powerful specific tools in mathematical analysis. 8), ([12], Proposition 3.9) This provides a first representation result, using a linear isomorphism in this case: A Banach space is representable as an order-continuous Banach function space with a weak unit if and only if it is the range of an integration map of a (countably additive) vector measure which is an isomorphism. This technique has been widely used for the identification of the optimal domain of some relevant operators; the reader can find some examples and applications in [12,13,14,15,16] and the references therein

Standard Definitions and Basic Concepts

The Basic Construction

Order-Continuous Banach Function Subspaces and pth Powers of Banach Spaces

Applications

Banach Function Subspaces of Multiplication Operators between Banach Spaces

Orthogonal Polynomials on Banach Spaces