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 ${\mathcal{R}}^n$ with decreasing components that satisfy the pointwise weak-majorization inequality $\lambda \left(|T\right(x\left)|\right)\underset{w}{\prec }q\ast \lambda \left(|x|\right)$, where λ is the eigenvalue map and * denotes the componentwise product in ${\mathcal{R}}^n$. 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 $T^{\ast }\left(e\right)$, 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.

Keywords:

• Euclidean Jordan algebra
• eigenvalue map
• weak-majorization ordering
• positive map
• spectral norm

AMS Subject Classifications:

• 15A42
• 17C20

Acknowledgments

We thank the referee for his/her comments and mentioning reference [16]. 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).

Additional information

Funding

The second author was financially supported by the National Research Foundation of Korea NRF-2016R1A5A1008055.