An infinite dimensional Schur–Horn Theorem and majorization theory

Abstract

The main result of this paper is the extension of the Schur–Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequences ξ and η that converge to 0, there exists a positive compact operator A with eigenvalue list η and diagonal sequence ξ if and only if ∑ j = 1 n ξ j ⩽ ∑ j = 1 n η j for every n if and only if ξ = Q η for some orthostochastic matrix Q. When ξ and η are summable, requiring additionally equality of their infinite series obtains the same conclusion, extending a theorem by Arveson and Kadison. Our proof depends on the construction and analysis of an infinite product of T-transform matrices.