Some log and weak majorization inequalities in Euclidean Jordan algebras

Abstract

Motivated by Horn's log-majorization (singular value) inequality and the related weak-majorization inequality for square complex matrices, we consider their Hermitian analogs for positive semidefinite matrices and for general (Hermitian) matrices, where denotes the Jordan product of A and B and denotes the componentwise product in . In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form for and for all a and b, where and denote, respectively, the quadratic representation and the eigenvalue vector of an element u. We also describe inequalities of the form , where A is a real symmetric positive semidefinite matrix and is the Schur product of A and b. In the form of an application, we prove the generalized Hölder type inequality , where denotes the spectral p-norm of x and with . We also give precise values of the norms of the Lyapunov transformation and relative to two spectral p-norms.