Group majorization, eaton triples and numerical range*

Abstract

Let G be a closed subgroup of the orthogonal group on a finite dimensional real inner product space V. The triple(V,G,F) is an Eaton triple if F≌V is a nonempty closed convex cone such that is nonempty for each x∈V. for all x,y∈F The Eaton triple (W,H,F)is called a reduced triple of the Eaton triple (V,G,F) if . In this event, by using a recent result of Niezgoda, a generaliszation of a result of Thompson and Poon on the convex hull of the G-orbit is given. Some matrix inequalities of Ky Fan on the sum of matrices are also extended. A generalized numerical range is introduced and we extend some results of Au-Yeung and Tsing on the equivalent statements of the convexity of the generalized numerical range associated with some Lie group. A question raised in [21]is answered by using a polygonal path result of Eaton and Perlman. We also discuss a possible extension of Schur-Horn-Kostant convexity result. Some questions are asked. The studies are motivated by some existing Lie theoretical results.