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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A noncompletely continuous operator on $L_{1}(G)$ whose random Fourier transform is in $c_{0}(\hat G)$
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by N. Ghoussoub and M. Talagrand PDF
Proc. Amer. Math. Soc. 92 (1984), 229-232 Request permission

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
  • Herman Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Statistics 23 (1952), 493–507. MR 57518, DOI 10.1214/aoms/1177729330
  • Hicham Fakhoury, Représentations d’opérateurs à valeurs dans $L^{1}(X,\,\Sigma ,\,\mu )$, Math. Ann. 240 (1979), no. 3, 203–212 (French). MR 526843, DOI 10.1007/BF01362310
  • N. Kalton, The endomorphisms of ${L_p}(0 \leqslant p \leqslant 1)$, preprint, 1979. H. P. Rosenthal, Convolution by a biased coin, The Altgeld book 1975/1976, University of Illinois Functional Analysis Seminar.
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • 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