The Kadison-Singer and Paulsen Problems in Finite Frame Theory

28 Peter G. CasazzaProposition 7. Let H M be an M-dimensional Hilbert space with orthonormalbasis {e i } M i=1 . Let P, Q be the orthogonal projections of HM onto N-dimensional subspaces W 1 , W 2 respectively. Then the chordal distance betweenW 1 , W 2 satisfiesd 2 c(W 1 , W 2 ) = 1 M∑‖P e i − Qe i ‖ 2 .2i=1In particular, there are orthonormal bases {e i } N i=1 for W 1 and {ẽ i } N i=1 for W 2satisfying1M∑N∑N∑‖P e i − Qe i ‖ 2 ≤ ‖e i − ẽ i ‖ 2 ≤ 2 ‖P e i − Qe i ‖ 2 .2i=1Proof. We compute:i=1i=1i=1M∑M∑‖P e i − Qe i ‖ 2 = 〈P e i − Qe i , P e i − Qe i 〉i=1M∑M∑M∑= ‖P e i ‖ 2 + ‖Qe i ‖ 2 − 2 〈P e i , Qe i 〉i=1= 2N − 2i=1M∑〈P Qe i , e i 〉i=1= 2N − 2T r P Q= 2N − 2[N − d 2 c(W 1 , W 2 )]= 2d 2 c(W 1 , W 2 ).This combined with Equation 1.5 completes the proof.The next problem to be addressed is to connect the distance betweenprojections and the distance between the corresponding ranges of analysisoperators for Parseval frames.Theorem 16. Let P, Q be projections of rank N on H M and let {e i } M i=1 bethe coordinate basis of H M . Further, assume that there is a Parseval frame{ϕ i } M i=1 for HN with analysis operator T satisfying T ϕ i = P e i for all i =1, 2, . . . , M. IfM∑‖P e i − Qe i ‖ 2 < ɛ,i=1then there is a Parseval frame {ψ i } M i=1 for H M with analysis operator T 1satisfyingT 1 ψ i = Qe i , for all i = 1, 2, . . . , M,andi=1