Infinite dimensional restricted invertibility

TL;DR

This paper extends the Bourgain-Tzafriri Restricted Invertibility Theorem to infinite-dimensional Hilbert spaces, introducing a new density concept and proving the theorem for specific classes of operators and frames.

Contribution

It provides a general definition of restricted invertibility in infinite dimensions using a new density notion, and proves the theorem for $\, ext{ell}_1$-localized operators and Gabor frames.

Findings

Localized Bessel systems contain large Riesz basic subsequences.

The strongest form of the infinite dimensional restricted invertibility theorem is established for certain operators.

A new density concept simplifies and broadens the application of the theorem.

Abstract

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.