The Horn Conjecture for Sums of Compact Selfadjoint Operators

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 Ho...

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Bibliographic Details
Published in:American journal of mathematics 2009-12, Vol.131 (6), p.1543-1567
Main Authors: Bercovici, H., Li, W. S., Timotin, D.
Format: Article
Language:English
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
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Summary: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.
ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.0.0085