Invertibility of ‘large’ submatrices with applications to the geometry of Banach spaces and harmonic analysis
Invertibility of ‘large’ submatrices with applications to the geometry of Banach spaces and harmonic analysis
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“大”子矩阵的可逆性及其在 Banach 空间几何和调和分析中的应用
DOI：
10.1007/bf02772174
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发表时间：
1987
影响因子：
1
通讯作者：
L. Tzafriri
中科院分区：
文献类型：
--
作者：
J. Bourgain;L. Tzafriri
摘要 本文研究的主要问题是作用于有限维lp空间的线性算子的受限可逆性。我们研究这些问题的最初动机在于它们的应用。下面获得的结果使我们能够完成具有极值欧几里得距离的Lp空间的补子空间结构的早期工作。设A是一个实数×n矩阵，被视为lpn上的线性算子； l≤p≤∞。通过 A 的受限可逆性，我们的意思是存在 {1, 2, …,n} 的子集 σ，使得 |σ|当限制于单位向量 eii=1n 的线性跨度时，∼n 和 A 充当同构。该性质在多种条件下成立。例如，如果 A 的范数 ‖A‖p 的边界是一个与 n 无关的常数，且 A 的对角线是单位矩阵，则存在一个索引集 σ, |σ| ∼n，其中（Rσ）有一个有界逆σ代表限制图）。这是通过简单地构造集合 σ 来实现的，使得••Rσ(A-I)R••p<21。p=2 的情况特别令人感兴趣。尽管该问题纯粹是希尔伯特问题，但证明除了空间l2 之外还涉及空间l1。这些方法是概率性的和组合性的。格洛腾迪克定理的关键用途。本文还对三角系统在正测度集上的行为进行了很好的应用，概括了谐波密度的结果。给定正勒贝格测度的圆 T 的子集 B，存在正密度 dens Λ > 0 的整数 Z 的子集 Λ，使得只要傅里叶变换的支持，{fx137-1}
$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} $$
关闭位于Λ。这里涉及的矩阵是洛朗矩阵。受限可逆性问题在 lp 空间类之外是有意义的，如单独的部分所示。然而，大多数论文都使用特定的 lp 技术，并且只有在 lp 空间的背景下才能获得完整的结果。
AbstractThe main problem investigated in this paper is that of restricted invertibility of linear operators acting on finite dimensionallp-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 ofLp-spaces which have extremal euclidean distance.LetA be a realn ×n matrix considered as a linear operator onlpn; 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 vectorseii=1n 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 ••Rσ(A-I)Rσ••p<21.The casep=2 is of particular interest. Although the problem is purely Hilbertian, the proofs involve besides the spacel2 also the spacel1. 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 oflp-spaces, as is shown in a separate section. However, most of the paper uses specificlp-techniques and complete results are obtained only in the context oflp-spaces.