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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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  • [1] W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces, North-Holland, Amsterdam, 1971.
  • [2] B. Z. Vulikh, Introduction to the theory of partially ordered spaces, Translated from the Russian by Leo F. Boron, with the editorial collaboration of Adriaan C. Zaanen and Kiyoshi Iséki, Wolters-Noordhoff Scientific Publications, Ltd., Groningen, 1967. MR 0224522
  • [3] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
  • [4] N. Aronszajn and P. Szeptycki, On general integral transformations, Math. Ann. 163 (1966), 127–154. MR 190799,
  • [5] J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR 0058896
  • [6] A. Ionescu Tulcea and C. Ionescu Tulcea, Topics in the theory of lifting, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 48, Springer-Verlag New York Inc., New York, 1969. MR 0276438
  • [7] W. A. J. Luxemburg and A. C. Zaanen, The linear modulus of an order bounded linear transformation. I, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math. 33 (1971), 422–434. MR 0303337
  • [8] A. C. Zaanen, Linear analysis, North-Holland, Amsterdam, 1964.

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