This is an announcement for the paper "Properties of representations of operators acting between spaces of vector-valued functions" by Delio Mugnolo and Robin Nittka.
Abstract: A well-known result going back to the 1930s states that all bounded linear operators mapping scalar-valued $L^1$-spaces into $L^\infty$-spaces are kernel operators and that in fact this relation induces an isometric isomorphism between those 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 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.
Archive classification: math.FA
Mathematics Subject Classification: 46G10, 47B34, 46M10, 47B65
Remarks: 13 pages
The source file(s), dunfordpettis.bbl: 5993 bytes
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http://arXiv.org/abs/0903.2038
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