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On subspaces generated by independent functions in symmetric spaces with the Kruglov property

Author: S. V. Astashkin
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 25 (2013), nomer 4.
Journal: St. Petersburg Math. J. 25 (2014), 513-527
MSC (2010): Primary 46E30
Published electronically: June 5, 2014
MathSciNet review: 3184613
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Abstract: For a broad class of symmetric spaces $ X$, it is shown that the subspace generated by independent functions $ f_k$ $ (k=1,2,\dots )$ is complemented in $ X$ if and only if so is the subspace in a certain symmetric space $ Z_X^2$ on the semiaxis generated by their disjoint shifts $ \bar {f}_k(t)=f_k(t-k+1)\chi _{[k-1,k)}(t)$. Moreover, if $ \sum _{k=1}^\infty m({\mathrm {supp}}f_k)\le 1$, then $ Z_X^2$ can be replaced by $ X$ itself in the last statement. This result is new even for $ L_p$-spaces. Some consequences are deduced; in particular, it is shown that symmetric spaces enjoy an analog of the well-known Dor-Starbird theorem on the complementability in $ L_p[0,1]$ $ (1\le p<\infty )$ of the closed linear span of some independent functions under the assumption that this closed linear span is isomorphic to $ \ell _p$.

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Additional Information

S. V. Astashkin
Affiliation: Samara State University 1, Academician Pavlov Street, 443011 Samara, Russia

Keywords: Complemented subspace, independent functions, Rademacher functions, symmetric space, Kruglov property, Boyd indices, lower $p$-estimate
Received by editor(s): October 10, 2012
Published electronically: June 5, 2014
Additional Notes: The author was supported in part by RFBR (grant nos. 10-01-00077 and 12-01-00198)
Article copyright: © Copyright 2014 American Mathematical Society