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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Authors: N. Ghoussoub and M. Talagrand
Journal: Proc. Amer. Math. Soc. 92 (1984), 229-232
MSC: Primary 46G99; Secondary 47B38
DOI: https://doi.org/10.1090/S0002-9939-1984-0754709-1
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 $\varepsilon > 0$.


References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1984 American Mathematical Society