The Schur-Horn theorem for unbounded operators with discrete spectrum

TL;DR

This paper extends the Schur-Horn theorem to unbounded self-adjoint operators with discrete spectrum in infinite-dimensional Hilbert spaces, providing a comprehensive characterization of their diagonals.

Contribution

It generalizes the finite-dimensional Schur-Horn theorem to a class of unbounded operators with discrete spectrum, linking to prior results for projections and compact operators.

Findings

Characterization of diagonals for unbounded operators with discrete spectrum

Extension of Schur-Horn theorem to infinite-dimensional setting

Majorization results for diagonals of unbounded operators

Abstract

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, Kaftal and Weiss' results for positive compact operators, and Bownik and Jasper's 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.