An Infinite Dimensional Schur-horn Theorem and Majorization Theory with Applications to Operator Ideals

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 Pn j=1 ξj ≤ Pn j=1 ηj for every n if and only if ξ = Qη 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. Further results on majorization for infinite sequences providing “intermediate” sequences generalize known results from the finite case. Majorization properties and invariance under various classes of stochastic matrices are then used to characterize arithmetic mean closed operator ideals.