Factorization of vector measures and their integration operators

José Rodríguez

Colloquium Mathematicae (2016)

  • Volume: 144, Issue: 1, page 115-125
  • ISSN: 0010-1354

Abstract

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Let X be a Banach space and ν a countably additive X-valued measure defined on a σ-algebra. We discuss some generation properties of the Banach space L¹(ν) and its connection with uniform Eberlein compacta. In this way, we provide a new proof that L¹(ν) is weakly compactly generated and embeds isomorphically into a Hilbert generated Banach space. The Davis-Figiel-Johnson-Pełczyński factorization of the integration operator I ν : L ¹ ( ν ) → X is also analyzed. As a result, we prove that if I ν is both completely continuous and Asplund, then ν has finite variation and L¹(ν) = L¹(|ν|) with equivalent norms.

How to cite

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José Rodríguez. "Factorization of vector measures and their integration operators." Colloquium Mathematicae 144.1 (2016): 115-125. <http://eudml.org/doc/284081>.

@article{JoséRodríguez2016,
abstract = {Let X be a Banach space and ν a countably additive X-valued measure defined on a σ-algebra. We discuss some generation properties of the Banach space L¹(ν) and its connection with uniform Eberlein compacta. In this way, we provide a new proof that L¹(ν) is weakly compactly generated and embeds isomorphically into a Hilbert generated Banach space. The Davis-Figiel-Johnson-Pełczyński factorization of the integration operator $I_\{ν\}: L¹(ν) → X$ is also analyzed. As a result, we prove that if $I_\{ν\}$ is both completely continuous and Asplund, then ν has finite variation and L¹(ν) = L¹(|ν|) with equivalent norms.},
author = {José Rodríguez},
journal = {Colloquium Mathematicae},
keywords = {vector measure; weakly compactly generated Banach space; Hilbert generated Banach space; uniform eberlein compact; integration operator; completely continuous operator; asplund operator},
language = {eng},
number = {1},
pages = {115-125},
title = {Factorization of vector measures and their integration operators},
url = {http://eudml.org/doc/284081},
volume = {144},
year = {2016},
}

TY - JOUR
AU - José Rodríguez
TI - Factorization of vector measures and their integration operators
JO - Colloquium Mathematicae
PY - 2016
VL - 144
IS - 1
SP - 115
EP - 125
AB - Let X be a Banach space and ν a countably additive X-valued measure defined on a σ-algebra. We discuss some generation properties of the Banach space L¹(ν) and its connection with uniform Eberlein compacta. In this way, we provide a new proof that L¹(ν) is weakly compactly generated and embeds isomorphically into a Hilbert generated Banach space. The Davis-Figiel-Johnson-Pełczyński factorization of the integration operator $I_{ν}: L¹(ν) → X$ is also analyzed. As a result, we prove that if $I_{ν}$ is both completely continuous and Asplund, then ν has finite variation and L¹(ν) = L¹(|ν|) with equivalent norms.
LA - eng
KW - vector measure; weakly compactly generated Banach space; Hilbert generated Banach space; uniform eberlein compact; integration operator; completely continuous operator; asplund operator
UR - http://eudml.org/doc/284081
ER -

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