Given a sequence of $m$-tuples $\{\Lambda_K\}_{K\subseteq[n]}$, how can I determine whether there exists an $m\times n$ matrix $\Phi$ such that $\Lambda_K$ is the spectrum of $\Phi_K\Phi_K^*$ for every $K\subseteq[n]$? (Here, $\Phi_K$ denotes the submatrix of $\Phi$ whose columns indexed by $K$.)

Whenever $A,B\subseteq[n]$ are disjoint, we have $\Phi_{A}\Phi_{A}^*+\Phi_{B}\Phi_{B}^*=\Phi_{A\sqcup B}\Phi_{A\sqcup B}^*$, and so the Schur-Horn inequalities necessarily apply to $(\Lambda_A,\Lambda_B,\Lambda_{A\sqcup B})$. Is this necessary condition also sufficient?

`$\Phi \mapsto \{$`

spectrum of`$\Phi_K\Phi_K^*\}_K$`

and look at its critical values. It's not differentiable, but only where the $\Phi_K \Phi_K^*$ have repeated eigenvalues, so assume you're staying away from there. Anyway I don't see any reason for the answer to be given by linear inequalities (like Schur-Horn). – Allen Knutson Jun 7 '13 at 3:24