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

Astashkin, S.

doi:10.1090/S1061-0022-2014-01303-9

On subspaces generated by independent functions in symmetric spaces with the Kruglov property
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by S. V. Astashkin
Translated by: S. Kislyakov

St. Petersburg Math. J. 25 (2014), 513-527

DOI: https://doi.org/10.1090/S1061-0022-2014-01303-9

Published electronically: June 5, 2014

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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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