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On the Structure of L1 of a Vector Measure via its Integration Operator

Authors
  • Calabuig, J. M.1
  • Rodríguez, J.2
  • Sánchez-Pérez, E. A.1
  • 1 Universidad Politécnica de Valencia, Instituto Universitario de Matemática Pura y Aplicada (IUMPA-UPV), Camino de Vera, s/n, Valencia, 46022, Spain , Valencia (Spain)
  • 2 Universidad de Murcia, Departamento de Matemática Aplicada, Facultad de Informática, Espinardo, Murcia, 30100, Spain , Espinardo (Spain)
Type
Published Article
Journal
Integral Equations and Operator Theory
Publisher
Birkhäuser-Verlag
Publication Date
Mar 25, 2009
Volume
64
Issue
1
Pages
21–33
Identifiers
DOI: 10.1007/s00020-009-1670-5
Source
Springer Nature
Keywords
License
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Abstract

Geometric and summability properties of the integration operator associated to a vector measure m can be translated in terms of structure properties of the space L1(m). In this paper we study the cases of the integration operator being: (i) p-concave on Lp(m), or (ii) positive p-summing on L1(m) (where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \leq p < \infty$$\end{document}). We prove that (i) is equivalent to saying that L1(m) contains continuously the Lp space of a (non-negative scalar) control measure for m. On the other hand, we show that (ii) holds if and only if L1(m) is order isomorphic to the L1 space of a non-negative scalar measure.

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