Functional Analysis and Its Applications

, Volume 47, Issue 2, pp 148–151 | Cite as

On complementability of subspaces in symmetric spaces with the Kruglov property

Brief Communications


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,∞). Moreover, if Σ k=1 m(supp f k ) ⩽ 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 ⩽ p < ) of the subspace [f k ] generated by independent functions provided that it is isomorphic to the space l p is obtained.

Key words

complemented subspace independent functions Rademacher functions symmetric space Kruglov property Boyd indices lower p-estimate 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II. Function spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1979.MATHGoogle Scholar
  2. [2]
    W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Mem. Amer. Math. Soc., 19: 217 (1979).MathSciNetGoogle Scholar
  3. [3]
    B. S. Kashin and A. A. Saakyan, Orthogonal Series, Amer. Math. Soc., Providence, RI, 1989.MATHGoogle Scholar
  4. [4]
    H. P. Rosenthal, Israel J. Math., 8 (1970), 273–303.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    W. B. Johnson and G. Schechtman, Ann. Probab., 17:2 (1989), 789–808.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    S. V. Astashkin and F. A. Sukochev, Zap. Nauchn. Sem. POMI, 345 (2007), 25–50; English transl.: J. Math. Sci. (N.Y.), 148:6 (2008), 795–809.Google Scholar
  7. [7]
    S. V. Astashkin and F. A. Sukochev, Uspekhi Mat. Nauk, 65:6 (2010), 3–86; English transl.: Russian Math. Surveys, 65:6 (2010), 1003-1081.MathSciNetCrossRefGoogle Scholar
  8. [8]
    M. Sh. Braverman, Independent Random Variables and Rearrangement Invariant Spaces, Cambridge Univ. Press, Cambridge, 1994.MATHCrossRefGoogle Scholar
  9. [9]
    V. M. Kruglov, Teor. Veroyatn. Primen., 15:2 (1970), 331–336; English transl.: Theory Probab. Appl., 15:2 (1970), 319-324.Google Scholar
  10. [10]
    S. G. Krein, Ju. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators, Transl. Math. Monogr., vol. 54, Amer. Math. Soc., Providence, RI, 1982.Google Scholar
  11. [11]
    C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, MA, 1988.MATHGoogle Scholar
  12. [12]
    V. A. Rodin and E. M. Semenov, Funkts. Anal. Prilozhen., 13:2 (1979), 91–92; English transl.: Functional Anal. Appl., 13:2 (1979), 150-151.MathSciNetMATHGoogle Scholar
  13. [13]
    F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Graduate Texts in Math., vol. 233, Springer-Verlag, New York, 2006.Google Scholar
  14. [14]
    J. Creekmore, Nederl. Akad. Wetensch. Indag. Math., 43:2 (1981), 145–152.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    S. J. Dilworth, in: Handbook of the Geometry of Banach Spaces, vol. 1, North-Holland, Amsterdam, 2001, 497–532.CrossRefGoogle Scholar
  16. [16]
    L. E. Dor and T. Starbird, Compositio Math., 39:2 (1979), 141–175.MathSciNetGoogle Scholar
  17. [17]
    L. E. Dor, Ann. of Math., 102:3 (1975), 463–474.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    F. L. Hernandez and E. M. Semenov, J. Funct. Anal., 169:1 (1999), 52–80.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Samara State UniversitySamaraRussia

Personalised recommendations