The Kadison-Singer and Paulsen Problems in Finite Frame Theory

16 Peter G. Casazza1.2.4 The Bourgain-Tzafriri ConjectureWe start with a fundamental theorem of Bourgain and Tzafriri called theRestricted Invertibility Principle. This theorem led to the (strong and weak)Bourgain-Tzafriri Conjectures. We will see that these conjectures are equivalentto PC.In 1987, Bourgain and Tzafriri [12] proved a fundamental result in Banachspace theory known as the Restricted Invertibility Principle.Theorem 8 (Bourgain-Tzafriri). There is a universal constants 0 < c < 1so that whenever T : l N 2 → l N 2 is a linear operator for which ‖T e i ‖ = 1,for 1 ≤ i ≤ N, then there exists a subset σ ⊂ {1, 2, . . . , N} of cardinality|σ| ≥ cN/‖T ‖ 2 so that for all choices of scalars {a j } j∈σ ,‖ ∑ a j T e j ‖ 2 ≥ c ∑j∈σj∈σ|a j | 2 .A close examination of the proof of the theorem [12] yields that c is onthe order of 10 −72 . The proof of the theorem uses probabilistic and functionanalytic techniques, is non-trivial and is non-constructive. A significant breakthroughoccurred recently when Spielman and Srivastava [39] presented analgorithm for proving the Restricted-Invertibility Theorem. Moreover, theirproof gives the best possible constants in the theorem.Theorem 9 (Restricted Invertibility Theorem: Spielman-SrivastavaForm). Assume {v i } M i=1 are vectors in lN 2 with A = ∑ Mi=1 v ivi T = I and0 < ɛ < 1. If L : l N 2 → l N 2 is a linear operator, then there is a subset J ⊂{1, 2, . . . , M} of size |J| ≥ ɛ 2 ‖L‖ 2 F‖L‖ 2and( )∑λ min Lv i (Lv i ) Ti∈Jfor which {Lv i } i∈J is linearly independent> (1 − ɛ)2 ‖L‖ FMwhere ‖L‖ F is the Frobenious norm of L and λ min is the smallest eigenvalueof the operator computed on span {v i } i∈J .This generalized form of the Restricted-Invertibility Theorem was introducedby Vershynin [41] where he studied the contact points of convex bodiesusing John’s decompositions of the identity. The corresponding theorem forinfinite dimensional Hilbert spaces is still open. But this case requires the setJ to be large with respect to the Beurling density [6, 7]. Special cases of thisproblem were solved in [24, 41].The inequality in the Restricted Invertibility Theorem is referred to as alower l 2 -bound. It is known [28, 41] that there is a corresponding close to oneupper l 2 -bound which can be achieved in the theorem. Recently, a two-sidedSpielman-Srivastava algorithm was given [25] which gives the best two sidedlower and upper bounds in BT.,