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The Horn conjecture for sums of compact selfadjoint operators
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 131, Number 6, December 2009
- pp. 1543-1567
- 10.1353/ajm.0.0085
- Article
- Additional Information
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 $\rho\ge1$.