Properties of Representations of Operators acting between Spaces of Vector-Valued Functions

Details Properties of Representations of Operators acting between Spaces of Vector-Valued Functions Properties of Representations of Operators acting between Spaces of Vector-Valued Functions

2009-03-11

arxiv.org/abs/0903.2038v2

Positivity 15 (2011), No. 1, 135-154

Mathematics

Functional Analysis

13 pages; we made the correspondence between regularity of the operator and the kernel more precise since the orginal version

Scientific paper

10.1007/s11117-010-0045-0

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 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.

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