On characteristics of the range of kernel operators

Gerlach, Moritz; Glück, Jochen

doi:10.1090/proc/16531

On characteristics of the range of kernel operators
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by Moritz Gerlach and Jochen Glück

Proc. Amer. Math. Soc. 152 (2024), 677-690

DOI: https://doi.org/10.1090/proc/16531

Published electronically: November 21, 2023

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Abstract:

We show that a positive operator between $L^p$-spaces is given by integration against a kernel function if and only if the image of each positive function has a lower semi-continuous representative with respect to a suitable topology. This is a consequence of a new characterization of kernel operators on general Banach lattices as those operators whose range can be represented over a fixed countable set of positive vectors. Similar results are shown to hold for operators that merely dominate a non-trivial kernel operator.

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