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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Estimation of a quadratic function and the $ p$-Banach-Saks property

Authors: E. M. Semenov and F. A. Sukochev
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 18 (2006), nomer 4.
Journal: St. Petersburg Math. J. 18 (2007), 647-656
MSC (2000): Primary 46E30
Published electronically: May 29, 2007
MathSciNet review: 2262587
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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}\Vert\sum_{1\le k\le m} y_n\Vert<\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 \lbrace1\rbrace$) that if $ \Gamma_i(E)\setminus\Gamma(E)\ne\varnothing$, then $ \Gamma_i(E)\setminus\Gamma(E)=\lbrace 2\rbrace$.

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Additional Information

E. M. Semenov
Affiliation: Voronezh State University, Universitetskaya Ploshchad′ 1, 394007 Voronezh, Russia

F. A. Sukochev
Affiliation: Flinders University of South Australia, Bedford Park, 5042, SA, Australia

Keywords: Rearrangement-invariant space, $p$-Banach--Saks property
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.
Article copyright: © Copyright 2007 American Mathematical Society