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On the representation of order continuous operators by random measures


Author: L. Weis
Journal: Trans. Amer. Math. Soc. 285 (1984), 535-563
MSC: Primary 47B38; Secondary 60G57
DOI: https://doi.org/10.1090/S0002-9947-1984-0752490-8
MathSciNet review: 752490
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Abstract: Using the representation $ Tf(y) = \smallint f\;d{v_y}$, where $ ({v_y})$ is a random measure, we characterize some interesting bands in the lattice of all order-continuous operators on a space of measurable functions. For example, an operator $ T$ is (lattice-)orthogonal to all integral operators (i.e. all $ {v_y}$ are singular) or belongs to the band generated by all Riesz homomorphisms (i.e. all $ {v_y}$ are atomic) if and only if $ T$ satisfies certain properties which are modeled after the Riesz homomorphism property [31] and continuity with respect to convergence in measure. On the other hand, for operators orthogonal to all Riesz homomorphisms (i.e. all $ {v_y}$ are diffuse) we give characterizations analogous to the characterizations of Dunford and Pettis, and Buhvalov for integral operators. The latter results are related to Enflo operators, to a result of J. Bourgain on Dunford-Pettis operators and martingale representations of operators.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0752490-8
Keywords: Positive operators in function spaces, transition kernels, integral operators, convergence in measure, vector-valued martingales
Article copyright: © Copyright 1984 American Mathematical Society

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