Factorization of vector measures and their integration operators

José Rodríguez¹

¹ Departamento de Matemática Aplicada Facultad de Informática Universidad de Murcia 30100 Espinardo (Murcia), Spain

Colloquium Mathematicum, Tome 144 (2016) no. 1, pp. 115-125.

Résumé

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

Détail

DOI : 10.4064/cm6735-11-2015

Keywords: banach space countably additive x valued measure defined sigma algebra discuss generation properties banach space its connection uniform eberlein compacta provide proof weakly compactly generated embeds isomorphically hilbert generated banach space davis figiel johnson czy ski factorization integration operator analyzed result prove completely continuous asplund has finite variation equivalent norms