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

TL;DR

This paper extends classical results on bounded linear operators between scalar-valued function spaces to vector-valued contexts, exploring kernel representations, local operators, and order properties in Banach space settings.

Contribution

It generalizes the isometric isomorphism between operators and kernels to vector-valued functions and extends the characterization of local operators to non-separable dual spaces.

Findings

Operators between vector-valued L^1 and L^∞ spaces are kernel operators.

Local operators on L^p spaces with values in non-separable dual Banach spaces are multiplication operators.

Positivity properties of operators correspond to properties of their representations.

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 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.