Estimation of a quadratic function and the $p$-Banach–Saks property
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E. M. Semenov and F. A. Sukochev
Translated by: A. Plotkin - St. Petersburg Math. J. 18 (2007), 647-656
- DOI: https://doi.org/10.1090/S1061-0022-07-00964-8
- Published electronically: May 29, 2007
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Abstract:
Let $E$ be a rearrangement-invariant Banach function space on $[0,1]$, and let $\Gamma (E)$ denote the set of all $p\ge 1$ such that any sequence $\lbrace x_n \rbrace$ in $E$ converging weakly to $0$ has a subsequence $\lbrace y_n \rbrace$ with $\sup _m m^{-1/p}\|\sum _{1\le k\le m} y_n\|<\infty$. The set $\Gamma _i(E)$ is defined similarly, but only sequences $\lbrace x_n \rbrace$ of independent random variables are taken into account. It is proved (under the assumption $\Gamma (E)\ne \lbrace 1\rbrace$) that if $\Gamma _i(E)\setminus \Gamma (E)\ne \varnothing$, then $\Gamma _i(E)\setminus \Gamma (E)=\lbrace 2\rbrace$.References
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Bibliographic Information
- E. M. Semenov
- Affiliation: Voronezh State University, Universitetskaya Ploshchad′ 1, 394007 Voronezh, Russia
- Email: semenov@func.vsu.ru
- F. A. Sukochev
- Affiliation: Flinders University of South Australia, Bedford Park, 5042, SA, Australia
- MR Author ID: 229620
- Email: sukochev@infoeng.flinders.edu.au
- Received by editor(s): February 22, 2006
- Published electronically: May 29, 2007
- Additional Notes: The first author was supported by RFBR (grant no. 05-01-00629) and by the “Universities of Russia” program (grant no. 04.01.051). The second author was supported by the Australia Research Counsil.
- © Copyright 2007 American Mathematical Society
- Journal: St. Petersburg Math. J. 18 (2007), 647-656
- MSC (2000): Primary 46E30
- DOI: https://doi.org/10.1090/S1061-0022-07-00964-8
- MathSciNet review: 2262587