The Schur-Horn Theorem for operators with finite spectrum

Abstract

We characterize the set of diagonals of the unitary orbit of a selfadjoint operator with a finite spectrum. Our result extends the Schur-Horn theorem from a finite dimensional setting to an infinite dimensional Hilbert space, analogous to Kadison's theorem for orthogonal projections, and the second author's result for operators with three point spectrum.

Diagonals of Self-adjoint Operators with Finite Spectrum

“…That is, we give a characterization of the diagonals of the set of all self-adjoint operators with a given spectrum. In the follow-up paper [3] we give a version of this result with prescribed multiplicities of eigenvalues.…”

“…, N n represent multiplicities of eigenvalues in the interior majorization inequality (1.5). In addition, the main result of [3] has much more complicated statement since it also involves exterior majorization conditions that are very sensitive to the locations of eigenvalues with infinite multiplicities. Hence, the multiplicity-free Theorem 1.2, though theoretically deductible from its counterpart, is not an easy consequence of [3].…”

“…The additional benefit of our approach is that it gives a short and largely independent proof, which does not rely on a long argument showing [3, Theorem 1.3] in its entirety. Instead, we merely use two particular results from [3] on the equivalence of Riemann and Lebesgue interior majorization and the sufficiency of Riemann majorization. Combining this with the key existence result in the non-summable case from [7] yields the sufficiency part of Theorem 1.2.…”

“…Note that Theorem 3.3 is concerned with numerical sequences without any mention to operators. The second result [3,Theorem 5.3] shows the existence of a self-adjoint operator with finite spectrum and prescribed diagonal under Riemann interior majorization. .…”

The Schur–Horn problem for normal operators

“…For the stateof-the-art results for von Neumann factors of type I ∞ , we direct the reader to the recent work of Kaftal and Weiss [14,15], Jasper [11], and Bownik and Jasper [7,8].…”