Some majorization inequalities induced by Schur products in Euclidean Jordan algebras

Abstract: In an Euclidean Jordan algebra V of rank n, an element x is said to be majorized by an element y, if the corresponding eigenvalue vector of x is majorized by the eigenvalue vector of y in R^n. In this article, we describe pointwise majorization inequalities of the form `T(x) majorized by S(x)', where T and S are linear transformations induced by Schur products. Specializing, we recover analogs of majorization inequalities of Schur, Hadamard, and Oppenheimer stated in the setting of Euclidean Jordan algebras, as well as majorization inequalities connecting quadratic and Lyapunov transformations on V. We also show how Schur products induced by certain scalar means (such as arithmetic, geometric, harmonic, and logarithmic means) naturally lead to majorization inequalities.