The Horn conjecture for compact selfadjoint operators

TL;DR

This paper extends Horn's inequalities to characterize eigenvalues of sums of compact selfadjoint operators, providing a complete set of inequalities including reverse versions, applicable even when only some eigenvalues are specified.

Contribution

It introduces the proper extension of Horn inequalities for infinite-dimensional operators and demonstrates their completeness in characterizing eigenvalues.

Findings

Extended Horn inequalities for compact selfadjoint operators.

Complete characterization of eigenvalues using these inequalities.

Applicable to cases with partial eigenvalue specifications.

Abstract

We determine the possible eigenvalues of compact selfadjoint operators A,B,C... with the property that A=B+C+... 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 A=B+C. 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 are specified. A special case is the requirement that B+C+... be positive of rank at most r.