The Kadison-Singer and Paulsen Problems in Finite Frame Theory

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12 Peter G. Casazzainto a finite number of subsets which were Riesz basic sequences. This led tothe conjecture:Feichtinger Conjecture 1 (FC) Every bounded frame (or equivalently,every unit norm frame) is a finite union of Riesz basic sequences.The finite dimensional form of FC looks like:Conjecture 7 (Finite Dimensional Feichtinger Conjecture). For every B, C >0, there is a natural number r = r(B, C) and a constant A = A(B, C) > 0so that whenever {ϕ i } N i=1 is a frame for HN with upper frame bound B and‖ϕ i ‖ ≥ C for all i = 1, 2, . . . , N, then {1, 2, . . . , N} can be partitioned intosubsets {A j } r j=1 so that for each 1 ≤ j ≤ r, {ϕ i} i∈Aj is a Riesz basic sequencewith lower Riesz basis bound A and upper Riesz basis bound B.There is a significant body of work on this conjecture [6, 7, 16, 31], yet, itremains open even for Gabor frames.We now check that the Feichtinger Conjecture is equivalent to PC.Theorem 5. The following are equivalent:(1) The Paving Conjecture.(2) The Feichtinger Conjecture.Proof. (1) ⇒ (2): Part (2) of Theorem 2 is equivalent to PC and clearlyimplies FC.(2) ⇒ (1): We will observe that FC implies Conjecture 4 which is equivalentto PC by Theorem 3. In Conjecture 4, {T e i } N i=1 is a frame for its spanwith upper frame bound 2. It is now immediate that the Finite DimensionalFeichtinger Conjecture implies Conjecture 4.Another equivalent formulation of KS due to Weaver [43].Conjecture 8 (KS r ). There are universal constants B and ɛ > 0 so that the followingholds. Let {ϕ i } M i=1 be elements of lN 2 with ‖ϕ i ‖ ≤ 1 for i = 1, 2, . . . , Mand suppose for every x ∈ l N 2 ,M∑|〈x, ϕ i 〉| 2 ≤ B‖x‖ 2 . (1.2)i=1Then, there is a partition {A j } r j=1 of {1, 2, . . . , M} so that for all x ∈ lN 2all j = 1, 2, . . . , r, ∑|〈x, ϕ i 〉| 2 ≤ (B − ɛ)‖x‖ 2 .i∈A jandTheorem 6. The following are equivalent:(1) The Paving Conjecture.(2) Conjecture KS r holds for some r ≥ 2.
1 The Kadison-Singer and Paulsen Problems in Finite Frame Theory 13Proof. Assume Conjecture KS r is true for some fixed r, B, ɛ. We will showthat Conjecture 1 is true. Let P be an orthogonal projection on H M withδ(P ) ≤ 1 B. If P has rank N, then its range is a N-dimensional subspace W ofH M . Define ϕ i = √ B · P e i ∈ W for all 1 ≤ i ≤ M. We check‖ϕ i ‖ 2 = B · ‖P e i ‖ 2 = B〈P e i , e i 〉 ≤ Bδ(P ) ≤ 1, for all i = 1, 2, . . . , M.Now, if x ∈ W is any unit vector thenM∑M∑|〈x, ϕ i 〉| 2 = |〈x, √ M∑BP e i 〉| 2 = B · |〈x, e i 〉| 2 = B.i=1i=1By Conjecture KS r , there is a partition {A j } r j=1 of {1, 2, . . . , M} satisfyingfor all 1 ≤ j ≤ r and all unit vectors x ∈ W ,∑|〈x, ϕ i 〉| 2 ≤ B − ɛ.i∈A jThen ∑ rj=1 Q A j= Id, and for any unit vector x ∈ W we havei=1i=1M∑M∑‖Q Aj P x‖ 2 = |〈Q Aj P x, e i 〉| 2 = |〈x, P Q j e i 〉| 2= 1 ∑|〈x, ϕ i 〉| 2 ≤ ɛ BB .i∈A jThus Conjecture 1 holds and so PC holds.Conversely, assume KS r fails for all r. Fix B = r ≥ 2 and let {ϕ i } M i=1in H N be a counterexample with ɛ = 1. Let ψ i = √ ϕiBand note that‖ψ i ψ T i ‖ = ‖ψ i ‖ 2 ≤ 1 B , for all i = 1, 2, . . . , M and ∑ Mi=1 ψ iψ T i ≤ Id. ThenId − ∑ Mi=1 ψ iψi T is a positive finite rank operator, so we can find positiverank one operators ψ i ψiT for M + 1 ≤ i ≤ K such that ‖ψ i ψi T ‖ ≤ 1 Bfor all1 ≤ i ≤ K and ∑ Ki=1 ψ iψi T = Id.Let T be the analysis operator for {ψ i } K i=1 , which is an isometry and if Pis the orthogonal projection of H K with range T (H N ), then P e i = T ψ i forall i = 1, 2, . . . , K. Let D be the diagonal matrix with the same diagonal asP . Then‖D‖ = max ‖ψ i‖ 2 ≤ 11≤i≤K B .Let {Q j } r j=1 be any K ×K diagonal projections which sum to the identity.Define a partition {A j } r j=1 of {1, 2, . . . , K by letting A j be the diagonal of Q j .By our choice of {ϕ i } M i=1 , there exists 1 ≤ j ≤ r and x ∈ HN with ‖x‖ = 1andi=1
Page 2 and 3: 2 Peter G. CasazzaThe Kadison-Singe
Page 4 and 5: 4 Peter G. Casazzaimplies PC follow
Page 6 and 7: 6 Peter G. Casazza(2) The square su
Page 8 and 9: 8 Peter G. CasazzaSimilarly, ‖
Page 10: 10 Peter G. Casazza‖N∑a i T e i
Page 15 and 16: 1 The Kadison-Singer and Paulsen Pr
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