Revisiting the Bourgain-tzafriri Restricted Invertibility Theorem

We will give some new techniques for working with problems surrounding the Bourgain-Tzafriri Restricted Invertibility Theorem. First we show that the parameters which work in the theorem for all ‖T‖ ≤ 2 √ 2 closely approximate the parameters which work for all operators. This yields a generalization of the theorem which simultaneously does restricted invertibility on a small partition of the vectors and yields a direct proof that the Bourgain-Tzafriri Conjecture is equivalent to the Feichtinger Conjecture. We also fill in two gaps in the theory involving the relationship between paving results for norm one operators with zero diagonal and restricted invertibility results.