Some inequalities involving determinants, eigenvalues, and Schur complements in Euclidean Jordan algebras

Abstract

In this paper, using Schur complements, we prove various inequalities in Euclidean Jordan algebras. Specifically, we study analogues of the inequalities of Fischer, Hadamard, Bergstrom, Oppenheim, and other inequalities related to determinants, eigenvalues, and Schur complements.

Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras

“…where E ij is the n × n matrix with 1s in the (i, j) and (j, i) slots and zeros elsewhere. It has been proved in [8], Theorem 8, that x∆y is a (symmetric) positive semidefinite matrix when x, y ≥ 0.…”

Some majorization inequalities in Euclidean Jordan algebras

“…By Theorem 3.1, for any a, b ∈ V , there exist Λ, Λ ∈ Aut(V ) such that |a + b| ≤ Λ(|a|) + Λ (|b|). This implies that λ ↓ i (|a + b|) ≤ λ ↓ i (Λ(|a|) + Λ (|b|)) for all i (see [6]). Since f is increasing function, we have f (λ ↓ i (|a + b|)) ≤ f (λ ↓ i (Λ(|a|) + Λ (|b|))) for all i.…”

Some majorization inequalities induced by Schur products in Euclidean Jordan algebras

“…This has been observed in [10] with a different proof and was crucially used in proving the Hölder type inequality [11] ||x • y|| 1 ≤ ||x|| p ||y|| q , where 1 ≤ p, q ≤ ∞ with 1 p + 1 q = 1. When x ≥ 0, a consequence of u + w ≺ x is the inequality det(x) ≤ det(u + v) = det(u) det(v) , which is known as the generalized Fischer inequality (see [13], Prop. 2).…”