Group majorization and Schur type inequalities

Niezgoda, Marek

doi:10.1016/s0024-3795(97)89322-6

Cited by 39 publications

(19 citation statements)

References 14 publications

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“…817-818]). The corresponding theory of group induced cone orderings was developed by Eaton and Perlman [7], Eaton [4,5,6], Giovagnoli and Wynn [8], Steerneman [18] and Niezgoda [16,17].…”

Section: Niezgodamentioning

confidence: 99%

“…It is known (see [16,Thm 3.2], [16, p. 14], [18,Thm 4.1] and [9]) that the following statements (i)-(iv) are mutually equivalent: (i) The group majorizations G and H are equivalent on W , that is…”

Section: Niezgodamentioning

confidence: 99%

“…Conditions (i)-(iv) of Section 2 are fulfilled (see [16,Example 4.1] for details). In particular, inequality (2.1) takes the form of the classical Schur inequality:…”

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confidence: 99%

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Upper bounds on certain functionals defined on groups of linear operators

Niezgoda¹

2006

ELA

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Abstract. The problem of estimating certain functionals defined on a group of linear operators generating a group induced cone (GIC) ordering is studied. A result of Berman and Plemmons [Math. Inequal. Appl., 2(1):149-152, 1998] is extended from the sum function to Schur-convex functions. It is shown that the problem has a closed connection with both Schur type inequality and weak group majorization. Some applications are given for matrices.Key words. Group majorization, GIC ordering, Normal decomposition system, Cone preordering, Schur type inequality, Schur-convex function, Eigenvalues, Singular values.AMS subject classifications. 06F20, 39B62, 15A48, 15A18. Introduction. Berman and Plemmons [2] proved that the functionalover all n × n orthogonal matrices U is maximized by an orthogonal matrix Q which simultaneously diagonalizes the symmetric matrices M j , j = 1, . . . , k. An analogous result holds for Hermitian matrices [2, Section 3].In the present paper we study a similar problem for a general linear space endowed with the structure of normal decomposition (ND) system (to be defined below). Also, we replace the sum function in (1.1) by an increasing function with respect to certain vector orderings (see Section 2). Some applications are given for matrices in Section 3. A further extension to weak group majorization is discussed in Section 4. Results.Let V be a finite-dimensional real linear space equipped with an inner product ·, · . By O(V ) we denote the orthogonal group acting on V . Let G be a closed subgroup of O(V ). The group majorization induced by G, abbreviated as G-majorization and written as G , is the preordering on V defined bywhere conv Gx denotes the convex hull of the orbit Gx := {gx : g ∈ G} (see [18]).We say that (V, G, (·) ↓ ) is a normal decomposition (ND) system (see [10,11]) if (A1) for any x ∈ V there exists g ∈ G satisfying x = gx ↓ , (A2) max g∈G x, gy = x ↓ , y ↓ for all x, y ∈ V . *

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“…817-818]). The corresponding theory of group induced cone orderings was developed by Eaton and Perlman [7], Eaton [4,5,6], Giovagnoli and Wynn [8], Steerneman [18] and Niezgoda [16,17].…”

Section: Niezgodamentioning

confidence: 99%

Section: Niezgodamentioning

confidence: 99%

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Upper bounds on certain functionals defined on groups of linear operators

Niezgoda¹

2006

ELA

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“…Indeed, all normal decomposition systems (E; G; ) with the group G ÿnite must have this form, by [26,Theorem 4:1]. The characterization problem for general groups G is studied in [22]. Proof.…”

Section: Group Invariant Convex Analysismentioning

confidence: 99%

Convex analysis on Cartan subspaces

Lewis

2000

Nonlinear Analysis: Theory, Methods & Applications

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“…where C G (x) = conv Gx denotes the convex hull of the orbit Gx = {gx: g ∈ G} (see [3][4][5]9,10]). It is known that…”

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confidence: 99%

On orthogonal projectors induced by compact groups and Haar measures

Niezgoda¹

2008

Journal of Mathematical Analysis and Applications

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We study the difference of two orthogonal projectors induced by compact groups of linear operators acting on a vector space. An upper bound for the difference is derived using the Haar measures of the groups. A particular attention is paid to finite groups. Some applications are given for complex matrices and unitarily invariant norms. Majorization inequalities of Fan and Hoffmann and of Causey are rediscovered. ResultsThroughout the section, V is a finite-dimensional real linear space with an inner product ·,· . The symbol O(V ) stands for the group of orthogonal operators acting on V . For a compact subgroup G of O(V ) we denote μ G = the Haar probability measure of G,It is known that P G is the orthogonal projector from V onto M G (see [11]). Theorem 1.1. Let H ⊂ K ⊂ O(V ) be compact groups such that μ K (H ) > 0. Denote by P and Q the orthogonal projectors from V onto M H and M K , respectively. Let ϕ : V → R be a K-invariant measurable positively homogeneous convex function such that ϕ(Then the following inequality holds:

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Group majorization and Schur type inequalities

Cited by 39 publications

References 14 publications

Upper bounds on certain functionals defined on groups of linear operators

Upper bounds on certain functionals defined on groups of linear operators

Convex analysis on Cartan subspaces

On orthogonal projectors induced by compact groups and Haar measures

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