Israel Journal of Mathematics

, 169:221 | Cite as

Classes of strictly singular operators and their products

  • G. Androulakis
  • P. Dodos
  • G. Sirotkin
  • V. G. Troitsky


V. D. Milman proved in [20] that the product of two strictly singular operators on L p [0, 1] (1 ⩽ p < 1) or on C[0, 1] is compact. In this note we utilize Schreier families \( \mathcal{S}_\xi \) in order to define the class of \( \mathcal{S}_\xi \)-strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying Banach space has finitely many equivalence classes of Schreier-spreading sequences. Finally we define the class of \( \mathcal{S}_\xi \)-hereditarily indecomposable Banach spaces and we examine the operators on them.


Banach Space Dimensional Subspace Basic Sequence Separable Banach Space Block Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [1]
    D. Alspach and S. A. Argyros, Complexity of weakly null sequences, Dissertationes Mathematicae 321 (1992).Google Scholar
  2. [2]
    G. Androulakis, E. Odell, Th. Schlumprecht and N. T. Jaegermann, On the structure of the spreading models of a Banach space, Canadian Journal of Mathematics 57 (2005), 673–707.zbMATHGoogle Scholar
  3. [3]
    S. A. Argyros and I. Deliyanni Examples of asymptotic ℓ 1 Banach spaces, Transactions of the American Mathematical Society 349 (1997), 973–995.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    S. A. Argyros, P. Dodos, Genericity and amalgamation of classes of Banach spaces, Advances in Mathematics 209 (2007), 666–748.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    S. A. Argyros, G. Godefroy, H. P. Rosenthal, Descriptive set theory and Banach spaces, in Handbook of the geometry of Banach spaces, vol. 2, North-Holland, Amsterdam, 2003, pp. 1007–1069.CrossRefGoogle Scholar
  6. [6]
    B. Beauzamy and J.-T. Lapreste, Modèles étalés des espaces de Banach, Travaux en Cours, Hermann, Paris, 1984.Google Scholar
  7. [7]
    C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces, Studia Mathematica 17 (1958), 151–164.zbMATHMathSciNetGoogle Scholar
  8. [8]
    B. Bossard, A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces, Fundamenta Mathematicae 172 (2002), 117–152.zbMATHMathSciNetGoogle Scholar
  9. [9]
    A. Brunel and L. Sucheston, On B convex Banach spaces, Mathematical Systems Theory 7 (1974), 294–299.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    A. Brunel and L. Sucheston, On J-convexity and some ergodic super properties of Banach spaces, Transaction of the American Mathematical Society 204 (1975), 79–90.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    J. Diestel, Sequences and Series in Banach Spaces, Springer Verlag, New York, 1984.Google Scholar
  12. [12]
    V. Ferenczi, Operators on subspaces of hereditarily indecomposable Banach spaces, Bulletin of the London Mathematical Society 29 (1997), 338–344.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    W. T. Gowers and B. Maurey, The unconditional basic sequence problem, Journal of the American Mathematical Society 6 (1993), 851–874.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    N. D. Hooker, Lomonosov’s hyperinvariant subspace theorem for real spaces, Mathematical Proceedings of the Cambridge Philosophical Society 89 (1981), 129–133.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    M. I. Kadets and A. Pelczyński, Bases, lacunary sequences and complemented subspaces in the spaces L p, Studia Mathematica 21 (1961/1962), 161–176.MathSciNetGoogle Scholar
  16. [16]
    A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics 156, Springer-Verlag, 1995.Google Scholar
  17. [17]
    J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I, Springer, Berlin, 1977.zbMATHGoogle Scholar
  18. [18]
    V. I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous operator, Academia Nauk SSSR. Funkcional’nyi Analiz i ego Priloženija 7 (1973), 55–56.MathSciNetGoogle Scholar
  19. [19]
    J. Lopez-Abad and A. Manoussakis, A classification of Tsirelson type spaces, preprint, arXiv:math. FA/0510410.Google Scholar
  20. [20]
    V. D. Milman, Operators of class C 0 and C* 0, Har’kovskii Ordena Trudovogo Krasnogo Znameni Universitet im. A. M. Gor’kogo. Teorija Funkcii Funkcional’nyi Analiz i ih Priloženija (1970), 15–26.Google Scholar
  21. [21]
    A. Plichko, Superstrictly singular and superstrictly cosingular operators, in Functional Analysis and its Applications, Elsevier, Amsterdam, 2004, pp. 239–255.CrossRefGoogle Scholar
  22. [22]
    A. Popov, Schreier singular operators, Houston Journal of Mathematics, to appear.Google Scholar
  23. [23]
    C. J. Read, A Banach space with, up to equivalence, precisely two symmetric bases, Israel Journal of Mathematics 40 (1981), 33–53.CrossRefMathSciNetGoogle Scholar
  24. [24]
    C. J. Read, Strictly singular operators and the invariant subspace problem, Studia Mathematica 132 (1991), 203–226.MathSciNetGoogle Scholar
  25. [25]
    H. P. Rosenthal, A characterization of Banach spaces containing l 1. Proceedings of the National Academy of Sciences of the United States of America 71 (1974), 2411–2413.zbMATHCrossRefGoogle Scholar
  26. [26]
    B. Sari, Th. Schlumprecht, N. Tomczak-Jaegermann and V. G. Troitsky, On norm closed ideals in L(ℓ p ⊗ ℓ q ), Studia Mathematica 179 (2007), 239–262.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    G. Sirotkin, A version of the Lomonosov invariant subspace theorem for real Banach spaces, Indiana University Mathematics Journal 54 (2005), 257–262.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2008

Authors and Affiliations

  • G. Androulakis
    • 1
  • P. Dodos
    • 2
  • G. Sirotkin
    • 3
  • V. G. Troitsky
    • 4
  1. 1.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  2. 2.Department of Mathematics, Faculty of Applied SciencesNational Technical University of AthensAthensGreece
  3. 3.Department of Mathematical SciencesNorthern Illinois UniversityDeKalbUSA
  4. 4.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

Personalised recommendations