Infinite dimensional restricted invertibility

The 1987 Bourgain–Tzafriri Restricted Invertibility Theorem is one of the most celebrated theorems in analysis. At the time of their work, the authors raised the question of a possible infinite dimensional version of the theorem. In this paper, we will give a general definition of restricted invertibility for operators on infinite dimensional Hilbert spaces based on the notion of density from frame theory. We then prove that localized Bessel systems have large subsets which are Riesz basic sequences. As a consequence, we prove the strongest possible form of the infinite dimensional restricted invertibility theorem for ℓ1-localized operators and for Gabor frames with generating function in the Feichtinger Algebra. For our exposition, we introduce a new notion of density which has serious advantages over the standard form because it is independent of index maps — and hence has much broader application.