Majorization for Infinite Sequences, an Extension of the Schur-horn Theorem, and Operator Ideals

نویسنده

  • VICTOR KAFTAL
چکیده

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 equality of the infinite series in the case that ξ and η 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.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

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 کامل

A 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 کامل

An infinite dimensional Schur–Horn Theorem and majorization theory

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, requirin...

متن کامل

The 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

The 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 کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2009