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

Abstract

Let $\mathcal{H}$ be a Hilbert space. Given a bounded positive definite operator $S$ on $\mathcal{H}$, and a bounded sequence $\mathbf{c}=\{c_k{\}}_{k\in \mathbb{N}}$ of non negative real numbers, the pair $(S,\mathbf{c})$ is frame admissible, if there exists a frame $\{f_k{\}}_{k\in \mathbb{N}}$ on $\mathcal{H}$ with frame operator $S$, such that $\Vert f_k{\Vert }^2=c_k$, $k\in \mathbb{N}$. 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 $(S,\mathbf{c})$, to be frame admissible.

To appear in Illinois Journal of Math