An infinite dimensional Schur–Horn Theorem and majorization theory
نویسندگان
چکیده
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 ∑n j=1 ξj ∑n 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. © 2010 Elsevier Inc. All rights reserved.
منابع مشابه
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, requir...
متن کاملA Survey on the Interplay between Arithmetic Mean Ideals, Traces, Lattices of Operator Ideals, and an Infinite Schur-horn Majorization Theorem
The main result in [24] on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals. In this survey we present the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sublattice structure of ideals, arithmetic mean idea...
متن کاملMajorization for Infinite Sequences, an Extension of the Schur-horn Theorem, and Operator Ideals
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 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. The similar result requiring equ...
متن کاملThe Schur-horn Theorem for Operators with Finite Spectrum
We characterize the set of diagonals of the unitary orbit of a self-adjoint operator with a finite spectrum. Our result extends the Schur-Horn theorem from a finite dimensional setting to an infinite dimensional Hilbert space, analogous to Kadison’s theorem for orthogonal projections [17, 18], and the second author’s result for operators with three point spectrum [16].
متن کاملThe Schur-horn Theorem for Operators and Frames with Prescribed Norms and Frame Operator
Let H be a Hilbert space. Given a bounded positive definite operator S on H, and a bounded sequence c = {ck}k∈N of non negative real numbers, the pair (S, c) is frame admissible, if there exists a frame {fk}k∈N on H with frame operator S, such that ‖fk‖ 2 = ck, k ∈ N. We relate the existence of such frames with the Schur-Horn theorem of majorization, and give a reformulation of the extended ver...
متن کامل