A Continuous Version of the Littlewood–Richardson Rule and Its Application to Invariant Subspaces

Bercovici, Hari; Li, Wing Suet; Smotzer, Thomas

doi:10.1006/aima.1997.1702

Cited by 7 publications

(6 citation statements)

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“…. , n. The hypothesis (2) implies that α (n) , β (n) , γ (n) satisfy the conditions of Theorem 2.1(2), and therefore there exist selfadjoint matrices A n , B n ∈ M n such that the eigenvalues of A n , B n , A n + B n are the components of α (n) , β (n) , γ (n) , respectively. Clearly, the norms of A n and B n are uniformly bounded, and therefore we can find integers n 1 < n 2 < · · · such that lim m→∞ p(A n m , B n m ) exists for every polynomial p in two noncommuting variables.…”

Section: Theorem Consider Bounded Nonincreasing Right-continuous Funmentioning

confidence: 94%

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Eigenvalue inequalities in an embeddable factor

Bercovici

2005

Proc. Amer. Math. Soc.

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Section: Theorem Consider Bounded Nonincreasing Right-continuous Funmentioning

confidence: 94%

“…Moreover, properties of the Littlewood-Richardson coefficients (cf., for instance, [6] or [2]) show that (I n , J n , K n ) belongs to T …”

Section: Theorem Consider Nonincreasing Sequences α β γ ∈ Rmentioning

confidence: 99%

Eigenvalue inequalities in an embeddable factor

Bercovici

2005

Proc. Amer. Math. Soc.

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“…One can for instance consider compact self-adjoint operators A, B, C on a Hilbert space and their eigenvalues. This analogue was considered by several authors [16,19], and a complete solution can be found in [5] for operators such that A, B, and −C are positive, and [6] for the general case. Without going into detail, let us say that these solutions are based on an understanding of the behavior of the Horn inequalities as the dimension of the space tends to infinity.…”

Section: S(e a I ) ∩ S(e B J ) ∩ S(e C K) = ∅mentioning

confidence: 95%

Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor

Bercovici

Collins

Dykema

et al. 2010

Journal of Functional Analysis

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The intersection ring of a complex Grassmann manifold is generated by Schubert varieties, and its structure is governed by the Littlewood-Richardson rule. Given three Schubert varieties S 1 , S 2 , S 3 with intersection number equal to one, we show how to construct an explicit element in their intersection. This element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags, and is therefore well defined over an arbitrary field of scalars. Moreover, this result also applies to appropriately defined analogues of Schubert varieties in the Grassmann manifolds associated with a finite von Neumann algebra. The arguments require the combinatorial structure of honeycombs, particularly the structure of the rigid extremal honeycombs. It is known that the eigenvalue distributions of self-adjoint elements a, b, c with a + b + c = 0 in the factor R ω are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. In particular, if x, y, z are self-adjoint elements of such a factor and x + y + z = 0, then there exist self-adjoint a, b, c ∈ R ω such that a + b + c = 0 and a (respectively, b, c) has the same eigenvalue distribution as x (respectively, y, z). A ('complete') matricial form of

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“…This way of labeling the Jordan blocks is more convenient for stating the Horn inequalities.) It has been known for some time [5,6,18] that the functions {θ n , θ ′ n , θ ′′ n : 1 ≤ n < ℵ 0 } satisfy a version of the Littlewood-Richardson rule provided that ∞ n=1 θ n ≡ 1 and, due to results of [15,16], this is equivalent (in the case of finite multiplicity N ) to saying that N n=1 θ n = N n=1 (θ ′ n θ ′′ n ), and that these functions satisfy a collection of divisibility relations, analogous to the Horn inequalities. The collection of these divisibility relations is indexed by triples of Schubert cells in a Grassmann variety…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Intersection theory and the Horn inequalities for invariant subspaces

Bercovici¹,

Li²

2016

Acta Sci. Math.

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Abstract. We provide a direct, intersection theoretic, argument that the Jordan models of an operator of class C 0 , of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where 'inequality' is replaced by 'divisibility'. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. The result is true even for operators acting on nonseparable Hilbert spaces. For such operators the usual Horn inequalities are supplemented so as to apply to all the Jordan blocks in the model.

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A Continuous Version of the Littlewood–Richardson Rule and Its Application to Invariant Subspaces

Cited by 7 publications

References 1 publication

Eigenvalue inequalities in an embeddable factor

Eigenvalue inequalities in an embeddable factor

Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor

Intersection theory and the Horn inequalities for invariant subspaces

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