The Horn conjecture for sums of compact selfadjoint operators

Abstract

We determine the possible nonzero eigenvalues of compact selfadjoint operators $A$, $B^{(1)}$, $B^{(2)}$, $\dots$, $B^{(m)}$ with the property that $A=B^{(1)}+B^{(2)}+\cdots+B^{(m)}$. 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 $B^{(1)}+B^{(2)}+\cdots+B^{(m)}$ be positive of rank at most $\rho\ge1$.