The Kadison-Singer and Paulsen Problems in Finite Frame Theory

6 Peter G. Casazza(2) The square sum of the coefficients of every row equals 1.The row vectors of the matrix B are not (δ, 2)-Rieszable, for any δ independentof N.Proof. A direct calculation yields (1) and (2).We will now show that the column vectors of B are not uniformly twoRieszable independent of N. So let {A 1 , A 2 } be a partition of {1, 2, . . . , 4N}.Without loss of generality, we may assume that |A 1 ∩ {1, 2, . . . , 2N}| ≥ N.Let the column vectors of the matrix B be {ϕ i } 4Ni=1 as elements of C2N . LetP N−1 be the orthogonal projection of C 2N onto the first N − 1 coordinates.Since |A 1 | ≥ N, there are scalars {a i } i∈A1 so that ∑ i∈A 1|a i | 2 = 1 andP N−1( ∑i∈A 1a i ϕ i)= 0.Also, let {ψ j } 2Nj=1 be the orthonormal basis consisting of the original columnsof the DF T 2N . We now have:‖ ∑a i ϕ i ‖ 2 = ‖(I − P N−1 )( ∑a i ϕ i )‖ 2i∈A 1 i∈A 1= 2N + 1 ‖(I − P N−1)( ∑≤ 2N + 1 ‖ ∑= 2N + 1= 2N + 1 .a i ψ i ‖ 2i∈A 1∑|a i | 2i∈A 1i∈A 1a i ψ i )‖ 2Letting N → ∞, this class of matrices is not (δ, 2)-Rieszable, and hence not(δ, 2)-pavable for any δ > 0.If this argument could be generalized to yield non-(δ, 3)-Rieszable (Pavable)matrices, then such an argument will lead to a complete counterexample toPC.1.2.2 The R ɛ -ConjectureIn this section we will define the R ɛ -Conjecture and show it is equivalent tothe Paving Conjecture.Definition 3. A family of vectors {ϕ i } M i=1 is an ɛ-Riesz basic sequence for0 < ɛ < 1 if for all scalars {a i } M i=1 we have