We determine the possible nonzero eigenvalues of compact selfadjoint operators A, with the property that A = B(1) + B(2) +…+ B(m). When all these operators are positive, the eigenvalues were known to be subject to certain inequalities which extend Horn's inequalities from the finite-dimensional case when m=2. We find the proper extension of the Horn inequalities and show that they, along with their reverse analogues, provide a complete characterization. Our results also allow us to discuss the more general situation where only some of the eigenvalues of the operators A and B(k) are specified. A special case is the requirement that B(1) + B(2) +…+ B(m) be positive of rank at most ρ ≥ 1.