Geodesic
On Complementability of Subspaces in Symmetric Spaces with the Kruglov Property
Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 2, pp. 80-84.

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We show that, for a broad class of symmetric spaces on $[0,1]$, the complementability of the subspace generated by independent functions $f_k$ $(k=1,2,\dots)$ 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. 
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S. V. Astashkin. On Complementability of Subspaces in Symmetric Spaces with the Kruglov Property. Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 2, pp. 80-84. http://geodesic.mathdoc.fr/item/FAA_2013_47_2_a7/

[1] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces II. Function spaces, Springer-Verlag, Berlin–Heidelberg–New York, 1979 | MR

[2] W. B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Mem. Amer. Math. Soc., 19:217 (1979) | MR

[3] B. S. Kashin, A. A. Saakyan, Ortogonalnye ryady, Izd-vo AFTs, M., 1999 | MR | Zbl

[4] H. P. Rosenthal, Israel J. Math., 8 (1970), 273–303 | DOI | MR | Zbl

[5] W. B. Johnson, G. Schechtman, Ann. Probab., 17:2 (1989), 789–808 | DOI | MR | Zbl

[6] S. V. Astashkin, F. A. Sukochev, Zap. nauchn. semin. POMI, 345 (2007), 25–50 | MR

[7] S. V. Astashkin, F. A. Sukochev, UMN, 65:6 (2010), 3–86 | DOI | MR

[8] M. Sh. Braverman, Independent Random Variables and Rearrangement Invariant Spaces, Cambridge Univ. Press, Cambridge, 1994 | MR | Zbl

[9] V. M. Kruglov, TVP, 15:2 (1970), 331–336 | MR

[10] S. G. Krein, Yu. I. Petunin, E. M. Semenov, Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR

[11] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, Boston, MA, 1988 | MR

[12] V. A. Rodin, E. M. Semenov, Funkts. analiz i ego pril., 13:2 (1979), 91–92 | MR | Zbl

[13] F. Albiac, N. J. Kalton, Topics in Banach Space Theory, Graduate Texts in Math., 233, Springer-Verlag, New York, 2006 | MR | Zbl

[14] J. Creekmore, Nederl. Akad. Wetensch. Indag. Math., 43:2 (1981), 145–152 | DOI | MR | Zbl

[15] S. J. Dilworth, Handbook of the geometry of Banach spaces, v. 1, North-Holland, Amsterdam, 2001, 497–532 | DOI | MR

[16] L. E. Dor, T. Starbird, Compositio Math., 39:2 (1979), 141–175 | MR

[17] L. E. Dor, Ann. of Math., 102:3 (1975), 463–474 | DOI | MR | Zbl

[18] F. L. Hernandez, E. M. Semenov, J. Funct. Anal., 169:1 (1999), 52–80 | DOI | MR | Zbl