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A lattice theoretic characterization of an integral operator

Author: Lawrence Lessner
Journal: Proc. Amer. Math. Soc. 53 (1975), 391-395
MSC: Primary 47B55
MathSciNet review: 0402533
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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)$.

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Keywords: Integral operator, lift, Riesz space
Article copyright: © Copyright 1975 American Mathematical Society

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