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.
Acknowledgments
We thank the referee for his/her comments and mentioning reference [Citation16]. The second author was financially supported by the National Research Foundation of Korea NRF-2016R1A5A1008055.
Disclosure statement
No potential conflict of interest was reported by the author(s).