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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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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Rademacher and Gaussian averages and Rademacher cotype of operators between Banach spaces
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Proc. Amer. Math. Soc. 128 (2000), 203-213 Request permission

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 \[ \left ( {\mathbf E} \| \sum _{k=1}^n Tx_k g_k\|^2 \right )^{1/2} \le c \left ( {\mathbf E} \| \sum _{k=1}^n x_k r_k\|^2 \right )^{1/2}, \] 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 \[ \varrho (T|{\mathcal G}_n,{\mathcal R}_n) = o(\sqrt {1+\log n}) \mbox {if and only if} \varrho (T|{\mathcal I}_n,{\mathcal R}_n) = o(\sqrt {n}).\]
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Additional Information
  • Aicke Hinrichs
  • Affiliation: Mathematical Institute, Friedrich-Schiller-University, D-07743 Jena, Germany
  • Email: nah@rz.uni-jena.de
  • 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
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 128 (2000), 203-213
  • MSC (1991): Primary 47D50, 46B07
  • DOI: https://doi.org/10.1090/S0002-9939-99-05012-1
  • MathSciNet review: 1621932