Abstract

Let mm be a vector measure taking values in a Banach space XX. We prove that if the integration operator Im:L1(m)→XI_m: L^1(m) \to X, Im(f)=∫f dmI_m(f)=\int f \, dm, is completely continuous and XX is Asplund, then mm has finite variation and L1(m)=L1(∣m∣)L^1(m) =L^1(|m|).

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Integration operator
  • vector measure
  • completely continuous operator
  • Asplund space

Mathematics Subject Classification

46E30, 46G10, 47B07

Suggested citation

J. M. Calabuig, J. Rodríguez, E. A. Sánchez-Pérez. “On Completely Continuous Integration Operators of a Vector Measure.” Journal of Convex Analysis 21 (2014), No. 3, 811–818.

Copyright Heldermann Verlag 2014