The Horn Conjecture for Sums of Compact Selfadjoint Operators - Bercovici, H.; Li, W. S.; Timotin, D. (American journal of mathematics, 2009)
We determine the possible nonzero eigenvalues of compact selfadjoint operators A,$B^{(1)} $,$B^{(2)} $, ...,$B^{(m)} $with the roperty that$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.