On Characteristics of the Range of Integral Operators

TL;DR

This paper characterizes when positive operators between L^p spaces are integral operators, linking their structure to lower semi-continuity and representing their range via a countable set of vectors.

Contribution

It provides a new characterization of kernel operators on Banach lattices based on their range representation over a countable set of vectors.

Findings

Positive operators are integral iff their images have lower semi-continuous representatives.

Kernel operators can be characterized by their range representation over countable sets.

Results extend to operators dominating non-trivial kernel operators.

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.