  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 quite 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 $\ell_1$-localized operators and for Gabor frames
with generating function in the Feichtinger Algebra. For our calculations, 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. We then show that in the setting of the restricted
invertibility theorem, this new density becomes equivalent to the standard
density.
