A pointwise weak-majorization inequality for linear maps over Euclidean Jordan algebras

Abstract

Given a linear map T on a Euclidean Jordan algebra of rank n, we consider the set of all nonnegative vectors q in with decreasing components that satisfy the pointwise weak-majorization inequality , where λ is the eigenvalue map and * denotes the componentwise product in . With respect to the weak-majorization ordering, we show the existence of the least vector in this set. When T is a positive map, the least vector is shown to be the join (in the weak-majorization order) of eigenvalue vectors of T(e) and , where e is the unit element of the algebra. These results are analogous to the results of Bapat [Majorization and singular values. III. Linear Algebra Appl. 1991;145:59–70] on singular values. We also extend two recent results of Tao et al. [Some log and weak majorization inequalities in Euclidean Jordan algebras. 2020. arXiv:2003.12377v2] proved for quadratic representations and Schur product induced transformations. As an application, we provide an estimate on the norm of a general linear map relative to spectral norms.