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Funktsional. Anal. i Prilozhen., 2013, Volume 47, Issue 2, Pages 80–84 (Mi faa3103)  

This article is cited in 1 scientific paper (total in 1 paper)

Brief communications

On Complementability of Subspaces in Symmetric Spaces with the Kruglov Property

S. V. Astashkin

Samara State University

Abstract: We show that, for a broad class of symmetric spaces on $[0,1]$, the complementability of the subspace generated by independent functions $f_k$ $(k=1,2,…)$ is equivalent to the complementability of the subspace generated by the disjoint translates $\bar{f}_k(t)=f_k(t-k+1)\chi_{[k-1,k)}(t)$ of these functions in some symmetric space $Z_X^2$ on the semiaxis $[0,\infty)$. Moreover, if $\sum_{k=1}^\infty m(\operatorname{supp}f_k)\le 1$, then $Z_X^2$ can be replaced by $X$ itself. This result is new even in the case of $L_p$-spaces. A series of consequences is obtained; in particular, for the class of symmetric spaces, a result similar to a well-known theorem of Dor and Starbird on the complementability in $L_p[0,1]$ $(1\le p<\infty)$ of the subspace $[f_k]$ generated by independent functions provided that it is isomorphic to the space $l_p$ is obtained.

Keywords: complemented subspace, independent functions, Rademacher functions, symmetric space, Kruglov property, Boyd indices, lower $p$-estimate

DOI: https://doi.org/10.4213/faa3103

Full text: PDF file (171 kB)
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English version:
Functional Analysis and Its Applications, 2013, 47:2, 148–151

Bibliographic databases:

UDC: 517.982.27
Received: 10.10.2011

Citation: S. V. Astashkin, “On Complementability of Subspaces in Symmetric Spaces with the Kruglov Property”, Funktsional. Anal. i Prilozhen., 47:2 (2013), 80–84; Funct. Anal. Appl., 47:2 (2013), 148–151

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. V. Astashkin, “On subspaces generated by independent functions in symmetric spaces with Kruglov property”, St. Petersburg Math. J., 25:4 (2014), 513–527  mathnet  crossref  mathscinet  zmath  isi  elib
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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