Positivity

, Volume 15, Issue 1, pp 135–154

Properties of representations of operators acting between spaces of vector-valued functions

Authors

  • Delio Mugnolo
    • Institute of AnalysisUniversity of Ulm
    • Institute of Applied AnalysisUniversity of Ulm
Article

DOI: 10.1007/s11117-010-0045-0

Cite this article as:
Mugnolo, D. & Nittka, R. Positivity (2011) 15: 135. doi:10.1007/s11117-010-0045-0

Abstract

A well-known result going back to the 1930s states that all bounded linear operators mapping scalar-valued L 1-spaces into L -spaces are kernel operators and that in fact this relation induces an isometric isomorphism between the space of such operators and the space of all bounded kernels. We extend this result to the case of spaces of vector-valued functions. A recent result due to Arendt and Thomaschewski states that the local operators acting on L p -spaces of functions with values in separable Banach spaces are precisely the multiplication operators. We extend this result to non-separable dual spaces. Moreover, we relate positivity and other order properties of the operators to corresponding properties of the representations.

Keywords

Vector measures Representation of operators Dunford–Pettis theorems Tensor products Kernel operators Multiplication operators Banach lattices Positive operators Regular operators p-tensor product

Mathematics Subject Classification (2000)

46G10 47B34 46M10 47B65

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010