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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A noncompletely continuous operator on $ L\sb{1}(G)$ whose random Fourier transform is in $ c\sb{0}(\hat G)$


Authors: N. Ghoussoub and M. Talagrand
Journal: Proc. Amer. Math. Soc. 92 (1984), 229-232
MSC: Primary 46G99; Secondary 47B38
MathSciNet review: 754709
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Abstract: Let $ T$ be a bounded linear operator from $ {L_1}(G,\lambda )$ into $ {L_1}(\Omega ,\mathcal{F},P)$, where $ (G,\lambda )$ is a compact abelian metric group with its Haar measure, and $ (\Omega ,\mathcal{F},P)$ is a probability space. Let $ ({\mu _\omega })$ be the random measure on $ G$ associated to $ T$; that is, $ Tf(\omega ) = \int_G {f(t)d{\mu _\omega }(t)} $ for each $ f$ in $ {L_1}(G)$.

We show that, unlike the ideals of representable and Kalton operators, there is no subideal $ B$ of $ \mathcal{M}(G)$ such that $ T$ is completely continuous if and only if $ {\mu _\omega } \in B$ for almost $ \omega $ in $ \Omega $. We actually exhibit a noncompletely continuous operator $ T$ such that $ {\hat \mu _\omega } \in {l_{2 + \varepsilon }}(\hat G)$ for each $ \varepsilon > 0$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0754709-1
PII: S 0002-9939(1984)0754709-1
Article copyright: © Copyright 1984 American Mathematical Society