Restricted invertibility of continuous matrix functions

TL;DR

This paper extends the concept of restricted invertibility to continuous matrix functions, establishing the existence of subspaces with uniform lower bounds on the norm of the matrix application, using orthogonality and probabilistic methods.

Contribution

It introduces new methods to find continuous subspaces with guaranteed invertibility bounds for matrix functions, improving understanding of their structure.

Findings

Existence of a continuous choice of subspaces with uniform lower bounds

Two methods: orthogonality and probabilistic, for constructing such subspaces

Optimal asymptotic dependence of subspace dimension on matrix size and norm

Abstract

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.