Diagonals of Self-adjoint Operators with Finite Spectrum

Given a finite set $X\subseteq \mathbb {R}$ 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 (2002) and the second author’s result for operators with three-point spectrum (2013).