The Kadison-Singer and Paulsen Problems in Finite Frame Theory

26 Peter G. CasazzaTheorem 15. Let Φ = {ϕ i } i∈I , Ψ = {ψ i } i∈I be Parseval frames for a Hilbertspace H with analysis operators T 1 , T 2 respectively. Ifd(Φ, Ψ) = ∑ i∈I‖ϕ i − ψ i ‖ 2 < ɛ,thend(T 1 (Φ), T 2 (Ψ)) = ∑ i∈I‖T 1 ϕ i − T 2 ψ i ‖ 2 < 4ɛ.Proof. Note that for all j ∈ I,T 1 ϕ j = ∑ i∈I〈ϕ j , ϕ i 〉e i , and T 2 ψ j = ∑ i∈I〈ψ j , ψ i 〉e i .Hence,‖T 1 ϕ j − T 2 ψ j ‖ 2 = ∑ |〈ϕ j , ϕ i 〉 − 〈ψ j , ψ i 〉| 2i∈I= ∑ |〈ϕ j , ϕ i − ψ i 〉 + 〈ϕ j − ψ j , ψ i 〉| 2i∈I≤ 2 ∑ |〈ϕ j , ϕ i − ψ i 〉| 2 + 2 ∑ |〈ϕ j − ψ j , ψ i 〉| 2 .i∈Ii∈ISumming over j and using the fact that our frames Φ and Ψ are Parsevalgives∑‖T 1 ϕ j − T 2 ψ j ‖ 2 ≤ 2 ∑ ∑|〈ϕ j , ϕ i − ψ i 〉| 2 + 2 ∑ ∑|〈ϕ j − ψ j , ψ i 〉| 2j∈Ij∈I i∈Ij∈I i∈I= 2 ∑ ∑|〈ϕ j , ϕ i − ψ i 〉| 2 + 2 ∑ ‖ϕ j − ψ j ‖ 2i∈I j∈Ij∈I= 2 ∑ ‖ϕ i − ψ i ‖ 2 + 2 ∑ ‖ϕ j − ψ j ‖ 2i∈Ij∈I= 4 ∑ ‖ϕ j − ψ j ‖ 2 .j∈INext, we want to relate the chordal distance between two subspaces to thedistance between their orthogonal projections. First we need to define thedistance between projections.Definition 6. If P, Q are projections on H N , we define the distance betweenthem byM∑d(P, Q) = ‖P e i − Qe i ‖ 2 ,i=1where {e i } N i=1 is an orthonormal basis for HN .