Journal of Convex Analysis 21 (2014), No. 3, 811--818
Copyright Heldermann Verlag 2014
On Completely Continuous Integration Operators of a Vector Measure
José M. Calabuig
Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain
jmcalabu@mat.upv.es
José Rodríguez
Departamento de Matemática Aplicada, Facultad de Informática, Universidad de Murcia, 30100 Espinardo (Murcia), Spain
joserr@um.es
Enrique A. Sánchez-Pérez
Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain
easancpe@mat.upv.es
[Abstract-pdf]
Let $m$ be a vector measure taking values in a Banach space $X$. We prove that if the integration operator $I_m: L^1(m) \to X$, $I_m(f)=\int f \, dm$, is completely continuous and $X$ is Asplund, then $m$ has finite variation and $L^1(m) =L^1(|m|)$.
Keywords: Integration operator, vector measure, completely continuous operator, Asplund space.
MSC: 46E30, 46G10, 47B07
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