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

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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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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Keywords

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