Upper bounds on certain functionals defined on groups of linear operators

Abstract

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 . *