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

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Abstract

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 n p ; 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 n i=1 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< 12 .

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.

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References

  1. K. I. Babenko,On conjugate functions, Dokl. Akad. Nauk SSSR62 (1948), 157–160 (Russian).

    MATH  MathSciNet  Google Scholar 

  2. G. Bennett,Probability inequalities for sums of independent random variables, J. Am. Stat. Assoc.57 (1962), 33–45.

    Article  MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Bourgain,New Classes of ℒ p -spaces, Lecture Notes in Math.889, Springer-Verlag, Berlin, 1981.

    Google Scholar 

  5. J. Bourgain,A remark on finite dimensional P λ-spaces, Studia Math.72 (1981), 87–91.

    MathSciNet  Google Scholar 

  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).

  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.

  8. J. Elton,Sign-embeddings of l 1n , Trans. Am. Math. Soc.279 (1983), 113–124.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  10. T. Figiel, J. Lindenstrauss and V. Milman,The dimension of almost spherical sections of convex bodies, Acta Math.139 (1977), 53–94.

    Article  MATH  MathSciNet  Google Scholar 

  11. T. Figiel, W. B. Johnson and G. Schechtman,Random sign-embeddings from l rn ; 2 <r < ∞, Proc. Am. Math. Soc., to appear.

  12. Y. Gordon,Some inequalities for Gaussian processes and applications, Isr. J. Math.50 (1985), 265–289.

    Article  MATH  Google Scholar 

  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.

  14. W. B. Johnson and L. Jones,Every L p operator is an L 2 operator, Proc. Am. Math. Soc.72 (1978), 309–312.

    Article  MATH  MathSciNet  Google Scholar 

  15. B. S. Kashin,Some properties of matrices bounded operators from space l n2 to l m2 , Izvestiya Akademii Nauk Armyanoskoi SSR, Matematika,15, No. 5 (1980), 379–394.

    MATH  MathSciNet  Google Scholar 

  16. Y. Katznelson and L. Tzafriri,On power bounded operators, J. Funct. Anal.68 (1986), 313–328.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. L. Krivine,Theoremes de factorisation dans les espaces reticules, Seminaire Maurey-Schwartz, 1973–74, Exposes 22–23, Ecole Polytech., Paris.

  18. S. Kwapien,Isomorphic characterisations of inner product spaces by orthogonal series with vector valued coefficients, Studia Math.44 (1972), 583–595.

    MATH  MathSciNet  Google Scholar 

  19. D. R. Lewis,Finite dimensional subspaces of L p , Studia Math.63 (1978), 207–212.

    MATH  MathSciNet  Google Scholar 

  20. J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces. I. Sequence Spaces, Springer-Verlag, Berlin, 1979.

    Google Scholar 

  21. J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces. II, Function Spaces, Springer-Verlag, Berlin, 1979.

    MATH  Google Scholar 

  22. B. Maurey,Theoremes de factorisations pour les operateures a valeurs dans un espace L p , Asterisque11, Soc. Math. France, 1974.

  23. A. Pajor,Sous espace l nl des espaces de Banach, Travaux En Cours, Hermann, Paris, 1985.

    Google Scholar 

  24. A. Pietsch,Absolute p-summierende Abbildugen in normierten Raumen, Studia Math.28 (1967), 333–353.

    MATH  MathSciNet  Google Scholar 

  25. W. Rudin,Trigonometric series with gaps, J. Math. Mech.9 (1960), 203–227.

    MATH  MathSciNet  Google Scholar 

  26. I. Z. Ruzsa,On difference sets, Studia Sci. Math. Hungar.13 (1978), 319–326.

    MATH  MathSciNet  Google Scholar 

  27. N. Sauer,On the density of families of sets, J. Comb. Theory Ser. A13 (1972), 145–147.

    Article  MATH  MathSciNet  Google Scholar 

  28. G. Schechtman,Fine embeddings of finite dimensional subspaces of L p , 1 ≦p < 2,into l 1 m , Proc. Am. Math. Soc.94 (1985), 617–623.

    Article  MATH  MathSciNet  Google Scholar 

  29. S. Shelah,A combinatorial problem: stability and order for models and theories in infinitary languages, Pacific J. Math.41 (1972), 247–261.

    MATH  MathSciNet  Google Scholar 

  30. N. Tomczak-Jaegermann,Computing 2-summing norm with few vectors, Ark. Mat.17 (1979), 173–177.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. IHES and University of Illinois at Urbana-Champaign, USA

    J. Bourgain

  2. The Hebrew University of Jerusalem, Israel

    L. Tzafriri

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  1. J. Bourgain
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  2. L. Tzafriri
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Bourgain, J., Tzafriri, L. Invertibility of ‘large’ submatrices with applications to the geometry of Banach spaces and harmonic analysis. Israel J. Math. 57, 137–224 (1987). https://doi.org/10.1007/BF02772174

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