Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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?)

  • 1. A. Hinrichs, On the type constants with respect to systems of characters of a compact abelian group, Stud. Math. 118 (1996), 231-243. MR 97i:46022
  • 2. S. Kwapie\'{n}, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Stud. Math. 44 (1972), 583-595. MR 49:5789
  • 3. M. Ledoux and M. Talagrand, Probability in Banach spaces, Springer, Berlin, Heidelberg, New York, 1991. MR 93c:60001
  • 4. B. Maurey and G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Stud. Math. 58 (1976), 45-90. MR 56:1388
  • 5. A. Pietsch and J. Wenzel, Orthonormal systems and Banach space geometry, Cambridge Univ. Press, 1998. CMP 99:01
  • 6. N. Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operator ideals, Longman, 1989. MR 90k:46039

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47D50, 46B07

Retrieve articles in all journals with MSC (1991): 47D50, 46B07

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