The Schur-Horn Problem for Normal Operators

TL;DR

This paper investigates the Schur-Horn problem for normal operators in infinite-dimensional von Neumann algebras, providing approximate solutions and extending classical finite-dimensional results to a broader context.

Contribution

It establishes an approximation theorem for the Schur-Horn problem in infinite-dimensional von Neumann algebras, generalizing Kadison's Carpenter Theorem.

Findings

Provided approximate solutions for the Schur-Horn problem in type I$_\infty$, II, and III von Neumann factors.

Developed an approximation theorem as an analogue of Kadison's Carpenter Theorem.

Extended the understanding of the Schur-Horn problem to infinite-dimensional operator algebras.

Abstract

We consider the Schur-Horn problem for normal operators in von Neumann algebras, which is the problem of characterizing the possible diagonal values of a given normal operator based on its spectral data. For normal matrices, this problem is well-known to be extremely difficult, and in fact, it remains open for matrices of size greater than $3$. We show that the infinite dimensional version of this problem is more tractable, and establish approximate solutions for normal operators in von Neumann factors of type I$_\infty$, II and III. A key result is an approximation theorem that can be seen as an approximate multivariate analogue of Kadison's Carpenter Theorem.