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 is also analyzed. As a result, we prove that if is both completely continuous and Asplund, then ν has finite variation and L¹(ν) = L¹(|ν|) with equivalent norms.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm6735-11-2015,
author = {Jos\'e Rodr\'\i guez},
title = {Factorization of vector measures and their integration operators},
journal = {Colloquium Mathematicae},
volume = {144},
year = {2016},
pages = {115-125},
zbl = {06574995},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm6735-11-2015}
}
José Rodríguez. Factorization of vector measures and their integration operators. Colloquium Mathematicae, Tome 144 (2016) pp. 115-125. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm6735-11-2015/