论文标题
连续矩阵函数的限制可逆性
Restricted invertibility of continuous matrix functions
论文作者
论文摘要
由Bourgain和Tzafriri的有影响力的结果激发，我们考虑连续矩阵函数$ a：\ Mathbb {r} \ to M_ {N \ Times n} $，以及较低的$ \ ell_2 $ -norm与它们与某些子集相关的限制。我们证明，对于任何具有单位长度列的$ a $，都有连续选择的子空间$ t \ mapsto u（t）\ subset \ subset \ mathbb {r}^n $，因此对于$ v \ in U（t）$，$ \ | a（t）$，$ \ | a（t）v \ | a（t）此外，选择$ u（t）$，以使其尺寸满足下界限，并具有对$ n $的最佳渐近依赖性和$ \ sup_ {t \ in \ Mathbb {r}}}} \ | a（t）\ |。我们提供两种方法。第一个依赖于正交性论点，而第二个是概率和合并性质的。后者不会产生$ \ dim（u（t））$的最佳限制，但是以这种方式获得的$ u（t）$保证具有典型的表示形式，因为由单位向量基础的子集跨越了连接的交叉空间。
Motivated by an influential result of Bourgain and Tzafriri, we consider continuous matrix functions $A:\mathbb{R}\to M_{n\times n}$ and lower $\ell_2$-norm bounds associated with their restriction to certain subspaces. We prove that for any such $A$ with unit-length columns, there exists a continuous choice of subspaces $t\mapsto U(t)\subset \mathbb{R}^n$ such that for $v\in U(t)$, $\|A(t)v\|\geq c\|v\|$ where $c$ is some universal constant. Furthermore, the $U(t)$ are chosen so that their dimension satisfies a lower bound with optimal asymptotic dependence on $n$ and $\sup_{t\in \mathbb{R}}\|A(t)\|.$ We provide two methods. The first relies on an orthogonality argument, while the second is probabilistic and combinatorial in nature. The latter does not yield the optimal bound for $\dim(U(t))$ but the $U(t)$ obtained in this way are guaranteed to have a canonical representation as joined-together spaces spanned by subsets of the unit vector basis.
