The Schur-Horn theorem for operators and frames with prescribed norms and frame operator

Abstract

Let H be a Hilbert space. Given a bounded positive definite operator S on H, and a bounded sequence c = {c k } k∈N of non negative real numbers, the pair (S, c) is frame admissible, if there exists a frame {f k } k∈N on H with frame operator S, such that f k 2 = c k , k ∈ N. We relate the existence of such frames with the Schur-Horn theorem of majorization, and give a reformulation of the extended version of Schur-Horn theorem, due to A. Neumann. We use it to get necessary conditions (and to generalize known sufficient conditions) for a pair (S, c), to be frame admissible.1991 Mathematics Subject Classification. Primary 42C15, Secondary 47A05.

Strong sums of projections in von Neumann factors

“…For instance if P is infinite but P = I , then there are no projections P and Q for which 1 5 5 and hence Q = I . But then, 1 5 P + 1 5 I = 2 5 P , whence P = P = I , against the assumption.…”

“…When A is also invertible, this is a special case of [1,Proposition 4.5]. In the case of compact operators, the diagonals of the partial isometry orbit are characterized in terms of majorization of sequences by the infinite dimensional Schur-Horn theorem obtained in [13].…”

“…Define 1] is flag, namely a monotone increasing strongly continuous net of projections with τ (E t ) = t for all t ∈ [0, 1] and A is not diagonalizable (in fact, it has no eigenvalues). Furthermore, 0 A 3 2 I 2I , R A = I , and it is easy to verify that …”

“…They proved that a sufficient condition for a positive bounded operator A ∈ B(H ) + to be the sum of projections is that its essential norm A e is larger than one [5,Theorem 2]. This result served as a basis for further work by Kornelson and Larson [16] and then by Antezana, Massey, Ruiz, and Stojanoff [1] on the decomposition of positive operators into (strongly converging) sums of rank-one positive operators of preset norms.…”

Minimization of convex functionals over frame operators

“…Section 3 is devoted to the basic facts about frames in C d together with some previous results from [3] about some design problems for frames. In Section 4, some properties of the convex functions P f defined on frame operators are given.…”