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

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Let H be a Hilbert space. Given a bounded positive definite operator S on H, and a bounded sequence c = {cₖ }ₖ ∈ ₙ of non negative real numbers, the pair (S, c) is frame admissible, if there exists a frame {fₖ }ₖ ∈ ₙ on H with frame operator S, such that fₖ ² = cₖ, 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.