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Rademacher and Gaussian averages
and Rademacher cotype of operators
between Banach spaces

Author: Aicke Hinrichs
Journal: Proc. Amer. Math. Soc. 128 (2000), 203-213
MSC (1991): Primary 47D50, 46B07
Published electronically: June 21, 1999
MathSciNet review: 1621932
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Abstract: A basic result of B. Maurey and G. Pisier states that Gaussian and Rademacher averages in a Banach space $X$ are equivalent if and only if $X$ has finite cotype. We complement this for linear bounded operators between Banach spaces. For $T\in {{\mathcal L}}(X,Y)$, let $\varrho(T|{\mathcal G}_n,{\mathcal R}_n)$ be the least $c$ such that

\begin{displaymath}\left( {\mathbf E} \| \sum _{k=1}^n Tx_k g_k\|^2 \right)^{1/2} \le c \left( {\mathbf E} \| \sum _{k=1}^n x_k r_k\|^2 \right)^{1/2}, \end{displaymath}

where ${\mathcal G}_n=(g_1,\ldots,g_n)$ and ${\mathcal R}_n=(r_1,\ldots,r_n)$ are systems of $n$ independent standard Gaussian and Rademacher variables, respectively. Let $\varrho(T|{\mathcal I}_n,{\mathcal R}_n)$ be the Rademacher cotype 2 norm of $T$ computed with $n$ vectors. We prove inequalities showing that the asymptotic behaviour of the sequence $\varrho(T|{\mathcal G}_n,{\mathcal R}_n)$ is almost determined by the asymptotic behaviour of the sequence $\varrho(T|{\mathcal I}_n,{\mathcal R}_n)$. In particular, we get

\begin{displaymath}\varrho(T|{\mathcal G}_n,{\mathcal R}_n) = o(\sqrt{1+\log n}) \ \ \mbox{if and only if} \ \ \varrho(T|{\mathcal I}_n,{\mathcal R}_n) = o(\sqrt{n}).\end{displaymath}

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

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Additional Information

Aicke Hinrichs
Affiliation: Mathematical Institute, Friedrich-Schiller-University, D-07743 Jena, Germany

Received by editor(s): June 5, 1997
Received by editor(s) in revised form: March 18, 1998
Published electronically: June 21, 1999
Additional Notes: The author is supported by DFG grant PI 322/1-1. The content of this paper is part of the author’s PhD-thesis written under the supervision of A. Pietsch.
Communicated by: Dale Alspach
Article copyright: © Copyright 1999 American Mathematical Society