A pointwise weak-majorization inequality for linear maps over Euclidean Jordan algebras
Loading...
Links to Files
Permanent Link
Author/Creator
Author/Creator ORCID
Date
2020-08-18
Type of Work
Department
Program
Citation of Original Publication
Muddappa Gowda and Jeong Juyoung, A pointwise weak-majorization inequality for linear maps over Euclidean Jordan algebras, https://arxiv.org/abs/2008.07472
Rights
This item is likely protected under Title 17 of the U.S. Copyright Law. Unless on a Creative Commons license, for uses protected by Copyright Law, contact the copyright holder or the author.
Subjects
Abstract
Given a linear map T on a Euclidean Jordan algebra of rank n, we consider the set of all nonnegative vectors q in Rⁿ with decreasing components that satisfy the pointwise weak-majorization inequality λ(|T(x)|)≺wq∗λ(|x|), where λ is the eigenvalue map and ∗ denotes the componentwise product in Rⁿ. 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∗(e), where e is the unit element of the algebra. These results are analogous to the results of Bapat, proved in the setting of the space of all n×n complex matrices with singular value map in place of the eigenvalue map. They also extend two recent results of Tao, Jeong, and Gowda 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.