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

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