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 nonnegative real numbers, the pair (S, c) is frame admissible, if there exists a frame {ƒ_k}_k ∈N on H with frame operator S, such that ||ƒ_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 this to get necessary conditions (and to generalize known sufficient conditions) for a pair (S, c) to be frame admissible.