Revista Matemática Complutense

, Volume 31, Issue 2, pp 351–377 | Cite as

Random unconditional convergence and divergence in Banach spaces close to \(L^1\)



We study conditions on Banach spaces close to \(L^1\) guaranteeing the existence of Random Unconditional Convergence and Divergence systems. Special attention is given to the Haar system and to Cesàro spaces.


Random unconditional convergence Schauder basis Haar functions Rearrangement invariant space Cesàro spaces 

Mathematics Subject Classification

Primary 46E30 46B15 Secondary 46B09 



The authors would like to thank Konstantin Lykov and Konstantin Tikhomirov for fruitful discussions at the early stages of this research. The first author acknowledges the support and hospitality of the Instituto de Matemáticas de la Universidad de Sevilla (IMUS).

We thank the referee for providing very useful suggestions.


  1. 1.
    Albiac, F., Kalton, N.J.: Topics in Banach Space Theory. Springer, New York (2006)MATHGoogle Scholar
  2. 2.
    Astashkin, S.V.: On the geometric properties of Cesàro spaces. Sb. Math. 203, 514–533 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Astashkin, S.V., Curbera, G.P., Tikhomirov, K.E.: On the existence of RUC systems in rearrangement invariant spaces. Math. Nach. 289, 175–186 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Astashkin, S.V., Maligranda, L.: Cesàro function spaces fail the fixed point property. Proc. Am. Math. Soc. 136, 4289–4294 (2008)CrossRefMATHGoogle Scholar
  5. 5.
    Astashkin, S.V., Maligranda, L.: Structure of Cesàro function spaces. Indag. Math. (N.S.) 20, 329–379 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)MATHGoogle Scholar
  7. 7.
    Billard, P., Kwapién, S., Pełczyński, A., Samuel, C.h.: Biorthogonal systems of random unconditional convergence in Banach spaces, In: Texas Functional Analysis Seminar 1985–1986, Longhorn Notes, pp. 13–35 (1986)Google Scholar
  8. 8.
    Curbera, G.P., Ricker, W.J.: Abstract Cesàro spaces: integral representations. J. Math. Anal. Appl. 441, 25–44 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dodds, P.G., Semenov, E.M., Sukochev, F.A.: RUC systems in rearrangement invariant spaces. Studia Math. 151, 161–173 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Garling, D.J.H., Tomczak-Jaegermann, N.: RUC-systems and Besselian systems in Banach spaces. Math. Proc. Camb. Philos. Soc. 106, 163–168 (1989)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Johnson, W.B., Maurey, B., Schechtman, G.: Weakly null sequences in \(L_1\). J. Am. Math. Soc. 20(1), 25–36 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kashin, B.S., Saakyan, A.A.: Orthogonal series. Am. Math. Soc., Providence RI (1989)Google Scholar
  13. 13.
    Krein, S.G., Petunin, Ju.I., Semenov, E.M.: Interpolation of Linear Operators (Am. Math. Soc., Providence RI) (1982)Google Scholar
  14. 14.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces, vol. II. Springer, Berlin (1979)CrossRefMATHGoogle Scholar
  15. 15.
    López-Abad, J., Tradacete, P.: Bases of random unconditional convergence in Banach spaces. Trans. Am. Math. Soc. 368, 9001–9032 (2016)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Novikov, I., Semenov, E.: Haar Series and Linear Operators. Kluwer, Dordrecht (1996)MATHGoogle Scholar
  17. 17.
    Ovsepian, R.I., Pełczyński, A.: On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space and related constructions of uniformly bounded orthogonal systems in \(L^2\). Studia Math. 54, 149–159 (1975)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Szarek, S.J.: On the best constants in the Khinchin inequality. Studia Math. 58(2), 197–208 (1976)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Wojtaszczyk, P.: Existence of some special bases in Banach spaces. Studia Math. 47, 83–93 (1973)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Wojtaszczyk, P.: Every separable Banach space containing \(c_0\) has an RUC system, Texas Functional Analysis Seminar 1985–1986, pp. 37–39, Longhorn Notes. Univ. Texas, Austin (1986)Google Scholar

Copyright information

© Universidad Complutense de Madrid 2017

Authors and Affiliations

  1. 1.Department of MathematicsSamara National Research UniversitySamaraRussia
  2. 2.Facultad de Matemáticas, Instituto de Matemáticas (IMUS)Universidad de SevillaSevillaSpain

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