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

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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.

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Authors and Affiliations

  1. Institute of Analysis, University of Ulm, 89069, Ulm, Germany

    Delio Mugnolo

  2. Institute of Applied Analysis, University of Ulm, 89069, Ulm, Germany

    Robin Nittka

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  1. Delio Mugnolo
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  2. Robin Nittka
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Correspondence to Robin Nittka.

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Mugnolo, D., Nittka, R. Properties of representations of operators acting between spaces of vector-valued functions. Positivity 15, 135–154 (2011). https://doi.org/10.1007/s11117-010-0045-0

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