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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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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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