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 $\varepsilon > 0$.

- 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 https://doi.org/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 https://doi.org/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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© Copyright 1984
American Mathematical Society