## A lattice theoretic characterization of an integral operator

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- by Lawrence Lessner
- Proc. Amer. Math. Soc.
**53**(1975), 391-395 - DOI: https://doi.org/10.1090/S0002-9939-1975-0402533-6
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## Abstract:

We are concerned here with obtaining necessary and sufficient conditions for a linear operator, $K:\mathcal {L}({{\text {X}}_1},\;{\mathcal {A}_1},\;{\mu _1}) \to M({{\text {X}}_2},\;{\mathcal {A}_2},\;{\mu _2})$, to be represented by an integral, $K(f) = \smallint k(x,\;y)f(y)\;dy$, with an ${\mathcal {A}_2} \times {\mathcal {A}_1}$ measurable kernel $k(x,\;y)$. That such conditions are developed in a lattice theoretic context will be shown to be quite natural. Our direction will be to characterize an integral operator by its action pointwise: i.e., $K()(x)$ is a linear functional on a subspace of the essentially bounded functions. Such a development leads one to define the kernel, $k(x,\;y)$, in a pointwise fashion also, and as a result we are confronted with the question of the ${\mathcal {A}_2} \times {\mathcal {A}_1}$ measurability of $k(x,\;y)$.## References

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## Bibliographic Information

- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**53**(1975), 391-395 - MSC: Primary 47B55
- DOI: https://doi.org/10.1090/S0002-9939-1975-0402533-6
- MathSciNet review: 0402533