Diagonals of self-adjoint operators

TL;DR

This paper extends the Schur-Horn theorem to infinite-dimensional self-adjoint operators, providing new proofs and inequalities relating their spectra and diagonals in advanced operator algebra contexts.

Contribution

It offers a new proof of the Schur-Horn theorem for positive trace-class operators on infinite-dimensional Hilbert spaces and establishes related inequalities in $II_1$ factors.

Findings

Generalization of Schur-Horn theorem to infinite dimensions

New proof techniques for operator diagonals

Inequalities linking spectral properties and diagonals in $II_1$ factors

Abstract

The eigenvalues of a self-adjoint nxn matrix A can be put into a decreasing sequence $\lambda=(\lambda_1,...,\lambda_n)$, with repetitions according to multiplicity, and the diagonal of A is a point of $R^n$ that bears some relation to $\lambda$. The Schur-Horn theorem characterizes that relation in terms of a system of linear inequalities. We give a new proof of the latter result for positive trace-class operators on infinite dimensional Hilbert spaces, generalizing results of one of us on the diagonals of projections. We also establish an appropriate counterpart of the Schur inequalities that relate spectral properties of self-adjoint operators in $II_1$ factors to their images under a conditional expectation onto a maximal abelian subalgebra.