Restricted invertibility of continuous matrix functions

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.