Israel Journal of Mathematics

, Volume 57, Issue 2, pp 137–224 | Cite as

Invertibility of ‘large’ submatrices with applications to the geometry of Banach spaces and harmonic analysis

  • J. Bourgain
  • L. Tzafriri


The main problem investigated in this paper is that of restricted invertibility of linear operators acting on finite dimensionall p -spaces. Our initial motivation to study such questions lies in their applications. The results obtained below enable us to complete earlier work on the structure of complemented subspaces ofL p -spaces which have extremal euclidean distance.

LetA be a realn ×n matrix considered as a linear operator onl p n ; l ≦p ≦ ∞. By restricted invertibility ofA, we mean the existence of a subset σ of {1, 2, …,n} such that |σ| ∼n andA acts as an isomorphism when restricted to the linear span of the unit vectorse i i=1 n There are various conditions under which this property holds. For instance, if the norm ‖A p ofA is bounded by a constant independent ofn and the diagonal ofA is the identity matrix, then there exists an index set σ, |σ| ∼n, for which (Rσ) has a bounded inverse σ stands for the restriction map). This is achieved by simply constructing the set σ so that ••R σ(A-I)R σ••p< 2 1 .

The casep=2 is of particular interest. Although the problem is purely Hilbertian, the proofs involve besides the spacel 2 also the spacel 1. The methods are probabilistic and combinatorial. Crucial use is made of Grothendieck’s theorem.

The paper also contains a nice application to the behavior of the trigonometric system on sets of positive measure, generalizing results on harmonic density. Given a subsetB of the circleT of positive Lebesgue measure, there exists a subset Λ of the integersZ of positive density dens Λ > 0 such that {fx137-1} whenever the support of the Fourier transform\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} \) off lies in Λ. The matrices involved here are Laurent matrices.

The problem of restricted invertibility is meaningful beyond the class ofl p -spaces, as is shown in a separate section. However, most of the paper uses specificl p -techniques and complete results are obtained only in the context ofl p -spaces.


Linear Operator Probability Space Independent Random Variable Linear Span Unconditional Basis 
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.


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  1. 1.
    K. I. Babenko,On conjugate functions, Dokl. Akad. Nauk SSSR62 (1948), 157–160 (Russian).zbMATHMathSciNetGoogle Scholar
  2. 2.
    G. Bennett,Probability inequalities for sums of independent random variables, J. Am. Stat. Assoc.57 (1962), 33–45.zbMATHCrossRefGoogle Scholar
  3. 3.
    G. Bennett, L. E. Dor, V. Goodman, W. B. Johnson and C. M. Newman,On uncomplemented subspaces of L p, 1 <p < 2, Isr. J. Math.26 (1977), 178–187.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    J. Bourgain,New Classes of ℒ p -spaces, Lecture Notes in Math.889, Springer-Verlag, Berlin, 1981.Google Scholar
  5. 5.
    J. Bourgain,A remark on finite dimensional P λ-spaces, Studia Math.72 (1981), 87–91.MathSciNetGoogle Scholar
  6. 6.
    J. Bourgain, P. G. Casazza, J. Lindenstrauss and L. Tzafriri,Banach spaces with a unique unconditional basis, up to permutation, Memoirs Am. Math. Soc.322 (1985).Google Scholar
  7. 7.
    S. Chevet,Series de variables aleatoires Gaussians a valeurs dans E \(\hat \otimes \) E F. Applications aux espaces de Wiener abstraits, Seminaire Maurey-Schwartz, 1977–78, Expose 19, Ecole Polytech., Paris.Google Scholar
  8. 8.
    J. Elton,Sign-embeddings of l n1, Trans. Am. Math. Soc.279 (1983), 113–124.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    T. Figiel and W. B. Johnson,Large subspaces of l n and estimates of the Gordon-Lewis constant, Isr. J. Math.37 (1980), 92–112.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    T. Figiel, J. Lindenstrauss and V. Milman,The dimension of almost spherical sections of convex bodies, Acta Math.139 (1977), 53–94.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    T. Figiel, W. B. Johnson and G. Schechtman,Random sign-embeddings from l nr; 2 <r < ∞, Proc. Am. Math. Soc., to appear.Google Scholar
  12. 12.
    Y. Gordon,Some inequalities for Gaussian processes and applications, Isr. J. Math.50 (1985), 265–289.zbMATHCrossRefGoogle Scholar
  13. 13.
    W. B. Johnson and G. Schechtman,On subspaces of L 1 with maximal distances to Euclidean space, Proc. Research Workshop on Banach Space Theory, University of Iowa (Bor-Luh-Lin, ed.), 1981, pp. 83–96.Google Scholar
  14. 14.
    W. B. Johnson and L. Jones,Every L p operator is an L 2 operator, Proc. Am. Math. Soc.72 (1978), 309–312.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    B. S. Kashin,Some properties of matrices bounded operators from space l 2n to l2m, Izvestiya Akademii Nauk Armyanoskoi SSR, Matematika,15, No. 5 (1980), 379–394.zbMATHMathSciNetGoogle Scholar
  16. 16.
    Y. Katznelson and L. Tzafriri,On power bounded operators, J. Funct. Anal.68 (1986), 313–328.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    J. L. Krivine,Theoremes de factorisation dans les espaces reticules, Seminaire Maurey-Schwartz, 1973–74, Exposes 22–23, Ecole Polytech., Paris.Google Scholar
  18. 18.
    S. Kwapien,Isomorphic characterisations of inner product spaces by orthogonal series with vector valued coefficients, Studia Math.44 (1972), 583–595.zbMATHMathSciNetGoogle Scholar
  19. 19.
    D. R. Lewis,Finite dimensional subspaces of L p, Studia Math.63 (1978), 207–212.zbMATHMathSciNetGoogle Scholar
  20. 20.
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces. I. Sequence Spaces, Springer-Verlag, Berlin, 1979.Google Scholar
  21. 21.
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces. II, Function Spaces, Springer-Verlag, Berlin, 1979.zbMATHGoogle Scholar
  22. 22.
    B. Maurey,Theoremes de factorisations pour les operateures a valeurs dans un espace L p, Asterisque11, Soc. Math. France, 1974.Google Scholar
  23. 23.
    A. Pajor,Sous espace l ln des espaces de Banach, Travaux En Cours, Hermann, Paris, 1985.Google Scholar
  24. 24.
    A. Pietsch,Absolute p-summierende Abbildugen in normierten Raumen, Studia Math.28 (1967), 333–353.zbMATHMathSciNetGoogle Scholar
  25. 25.
    W. Rudin,Trigonometric series with gaps, J. Math. Mech.9 (1960), 203–227.zbMATHMathSciNetGoogle Scholar
  26. 26.
    I. Z. Ruzsa,On difference sets, Studia Sci. Math. Hungar.13 (1978), 319–326.zbMATHMathSciNetGoogle Scholar
  27. 27.
    N. Sauer,On the density of families of sets, J. Comb. Theory Ser. A13 (1972), 145–147.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    G. Schechtman,Fine embeddings of finite dimensional subspaces of L p, 1 ≦p < 2,into l m1, Proc. Am. Math. Soc.94 (1985), 617–623.zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    S. Shelah,A combinatorial problem: stability and order for models and theories in infinitary languages, Pacific J. Math.41 (1972), 247–261.zbMATHMathSciNetGoogle Scholar
  30. 30.
    N. Tomczak-Jaegermann,Computing 2-summing norm with few vectors, Ark. Mat.17 (1979), 173–177.CrossRefMathSciNetGoogle Scholar
  31. 31.
    V. N. Vapnik and A. Ya. Cervonenkis,On uniform convergence of the frequencies of events to their probabilities, SIAM Theory of Prob. and Its Appl.16 (1971), 264–280.CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1987

Authors and Affiliations

  • J. Bourgain
    • 1
  • L. Tzafriri
    • 2
  1. 1.IHES and University of Illinois at Urbana-ChampaignUSA
  2. 2.The Hebrew University of JerusalemIsrael

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