The Horn Conjecture for Sums of Compact Selfadjoint Operators

• Published in: American journal of mathematics Vol. 131; no. 6; pp. 1543 - 1567
• Main Authors: Bercovici, H., Li, W. S., Timotin, D.
• Format: Journal Article
• Language: English
• Published: Baltimore, MD Johns Hopkins University Press 01.12.2009
• Subjects: Adjoints; Cardinality; Decreasing sequences; Eigenvalues; Exact sciences and technology; General mathematics; General, history and biography; Integers; Linear algebra; Mathematical analysis; Mathematical problems; Mathematical sequences; Mathematical vectors; Mathematics; Matrices; Nonlinear algebraic and transcendental equations; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Operator theory; Sciences and techniques of general use; Theorems; Compact operator; Eigenvalue; Mathematics; Rank; Self adjoint operator; Positive operator; Inequality
• ISSN: 0002-9327; 1080-6377; 1080-6377
• DOI: 10.1353/ajm.0.0085

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.