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 nonnegative 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 $\|f_k \|^2 = c_k$, $k \in \mathbb {N}$. 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 this to get necessary conditions (and
to generalize known sufficient conditions) for a pair $(S,
\mathbf{c})$ to be frame admissible.