  The main result of this paper is the extension of the Schur-Horn Theorem to
infinite sequences: For two nonincreasing nonsummable sequences x and y that
converge to 0, there exists a compact operator A with eigenvalue list y and
diagonal sequence x if and only if y majorizes x (\sum_{j=1}^n x_j \le
\sum_{j=1}^n y_j for all n) if and only if x = Qy for some orthostochastic
matrix Q. The similar result requiring equality of the infinite series in the
case that the sequences x and y are summable is an extension of a recent
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.
