Diagonals of self-adjoint operators with finite spectrum

TL;DR

This paper extends classical theorems to characterize diagonals of self-adjoint operators with finite spectrum in infinite-dimensional spaces, broadening understanding of spectral properties.

Contribution

It generalizes the Schur-Horn theorem to infinite-dimensional Hilbert spaces for operators with finite spectrum, building on Kadison's and previous results.

Findings

Characterization of diagonals for self-adjoint operators with finite spectrum

Extension of Schur-Horn theorem to infinite dimensions

Connection to Kadison's theorem for projections

Abstract

Given a finite set $X\subseteq\R$ 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 and the second author's result for operators with three point spectrum.