An Infinite Dimensional Version of the Schur–Horn Convexity Theorem

Neumann, Andreas

doi:10.1006/jfan.1998.3348

Cited by 57 publications

(99 citation statements)

References 3 publications

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“…But there is a fundamental difference in the nature of the characterizations of ref. 10 and the results below that goes beyond the fact that Neumann confines attention to self-adjoint operators. The comparison is clearly seen for the two-point set X ϭ {0, 1}.…”

Section: In This Paper We Address the Problem Of Determining The Elemmentioning

confidence: 88%

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Diagonals of normal operators with finite spectrum

Arveson

2007

Proc. Natl. Acad. Sci. U.S.A.

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Let X ‫؍‬ {1, . . . , N} be a finite set of complex numbers, and let A be a normal operator with spectrum X that acts on a separable Hilbert space H. Relative to a fixed orthonormal basis e 1, e2, . . . for H, A gives rise to a matrix whose diagonal is a sequence d ϭ (d1, d 2, . . . ) with the property that each of its terms dn belongs to the convex hull of X. Not all sequences with that property can arise as the diagonal of a normal operator with spectrum X. The case where X is a set of real numbers has received a great deal of attention over the years and is reasonably well (though incompletely) understood. In this work we take up the case in which X is the set of vertices of a convex polygon in ‫.ރ‬ The critical sequences d turn out to be those that accumulate rapidly in X in the sense that operator algebras ͉ Schur-Horn theorem ͉ spectral theory G iven a self-adjoint n ϫ n matrix A, the diagonal of A and the eigenvalue list of A are two points of ‫ޒ‬ n that bear some relation to each other. The Schur-Horn theorem characterizes that relation in terms of a system of linear inequalities (1, 2). That characterization has attracted a great deal of interest over the years and has been generalized in remarkable ways. For example, refs. 3-6 represent some of the milestones. More recently, a characterization of the diagonals of projections acting on infinite dimensional Hilbert spaces has been discovered (7,8), and a version of the Schur-Horn theorem for positive trace-class operators is given in ref. 9. The latter reference contains a somewhat more complete historical discussion.Let X be a finite subset of the complex plane ‫,ރ‬ and consider the set N(X) of all normal operators acting on a separable Hilbert space H that have spectrum X with uniformly infinite multiplicity,The set N(X) is invariant under the action of the group of *-automorphisms of B(H), and it is closed in the operator norm. Fixing an orthonormal basis e 1 , e 2 , . . . for H, one may consider the (nonclosed) set D(X) of all diagonals of operators in N(X) D͑X͒ ϭ ͕͑͗Ae 1 , e 1 ͘, ͗Ae 2 , e 2 ͘, . . . ͒͒ ʦ ᐉ ϱ : A ʦ N͑X͒}.In this paper we address the problem of determining the elements of D(X)., each term d n must belong to the convex hull of X. Indeed, since there is a normal operator A with spectrum X such that d n ϭ ͗Ae n , e n ͘, n Ն 1, each d n must belong to the numerical range of A, and the closure of the numerical range of a normal operator is the convex hull of its spectrum. This necessary condition d n ʦ conv X, n Ն 1, is not sufficient. Indeed, a characterization of D({0, 1}) (the set of diagonals of projections) was given by Kadison (8), the main assertion of which can be paraphrased as follows:Theorem 1 (Theorem 15 of ref. 8). Let d ϭ (d 1 , d 2 , . . . ) ʦ ᐉ ϱ be a sequence satisfying 0 Յ d n Յ 1 for every n andThen one has the following dichotomy:In a recent paper (9), a related spectral characterization was found for the possible diagonals of positive trace-class operators. That paper did not address the case of more general self-adjo...

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Section: In This Paper We Address the Problem Of Determining The Elemmentioning

confidence: 88%

“…In that case, the results of ref. 10 provide the following description of the closure of D(X) in the ᐉ ϱ -norm:…”

Section: In This Paper We Address the Problem Of Determining The Elemmentioning

confidence: 99%

Diagonals of normal operators with finite spectrum

Arveson

2007

Proc. Natl. Acad. Sci. U.S.A.

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“…This reformulation does not require an ordering of sequences. In [12] Neumann gave an infinite dimensional generalization of the Schur-Horn theorem in terms of ∞ -closure of the convexity condition (1.2). This gives a nice analogue of the finite dimensional theorem, but a great deal of information is lost in taking the closures.…”

Section: Theorem 11 (Schur-horn Theorem) Let {λmentioning

confidence: 99%

Diagonals of Self-adjoint Operators with Finite Spectrum

Bownik

Jasper

2015

Bull. Polish Acad. Sci. Math.

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“…This suggests that one could, in principle, use this tool in dealing with convexity problems such as those in the papers of Bloch, Flaschka, and Ratiu [4] or of Neumann [31]. The implementation of this idea is not free of difficulties and remains an open problem.…”

Section: Theorem Let F : X → V Be a Continuous Map From A Connected mentioning

confidence: 99%

Openness and convexity for momentum maps

BIRTEA

Ortega

Raţiu

2008

Trans. Amer. Math. Soc.

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Abstract. The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of a topological nature; more specifically, we show that convexity follows if the map is open onto its image and has the so-called local convexity data property. These conditions are satisfied in all the classical convexity theorems and hence they can, in principle, be obtained as corollaries of a more general theorem that has only these two hypotheses. We also prove a generalization of the socalled Local-to-Global Principle that only requires the map to be closed and to have a normal topological space as domain, instead of using a properness condition. This allows us to generalize the Flaschka-Ratiu convexity theorem to noncompact manifolds.

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An Infinite Dimensional Version of the Schur–Horn Convexity Theorem

Cited by 57 publications

References 3 publications

Diagonals of normal operators with finite spectrum

Diagonals of normal operators with finite spectrum

Diagonals of Self-adjoint Operators with Finite Spectrum

Openness and convexity for momentum maps

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