Diagonals of Self-adjoint Operators with Finite Spectrum

Abstract

Abstract. Given a finite set X ⊆ R we characterize the diagonals of self-adjoint operators with spectrum X. 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 [8,9] and the second author's result for operators with three point spectrum [7].

The Schur–Horn theorem for unbounded operators with discrete spectrum

“…

We characterize diagonals of unbounded self-adjoint operators on a Hilbert space H that have only discrete spectrum, i.e., with empty essential 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 [17,18], Kaftal and Weiss [21] results for positive compact operators, and Bownik and Jasper [10,11,16] characterization for operators with finite spectrum. Furthermore, we show that if a symmetric unbounded operator E on H has a nondecreasing unbounded diagonal, then any sequence that weakly majorizes this diagonal is also a diagonal of E.

…”

“…The Schur-Horn theorem has been extended to compact positive operators by Kaftal and Weiss [21] and Loreaux and Weiss [27] in terms of majorization inequalities [20]. Lebesgue type majorization was used by Bownik and Jasper [10,11,16] to characterize diagonals of self-adjoint operators with finite spectrum operators. Other notable progress includes the work of Arveson [5] on diagonals of normal operators with finite spectrum and Antezana, Massey, Ruiz, and Stojanoff's results [1].…”

The Schur-Horn Theorem for operators with finite spectrum

“…The case when the multiplicities of eigenvalues are not prescribed was already considered by the authors in [9]. The case when the multiplicities of eigenvalues are not prescribed was already considered by the authors in [9].…”

The Schur–Horn theorem for operators with three point spectrum

“…The case when the multiplicities of eigenvalues are not prescribed was already considered by the authors in [8]. While the main result in [8] provides a satisfactory description of possible diagonals of operators with finite spectrum, it is far from describing diagonals of the unitary orbit of such operators. In other words, a fully satisfactory Schur-Horn theorem should characterize the diagonals of operators with given eigenvalues and their corresponding multiplicities.…”