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

TL;DR

This paper establishes a weak-majorization inequality for linear maps on Euclidean Jordan algebras, identifying the least majorizing vector and extending classical matrix inequalities to a broader algebraic setting.

Contribution

It introduces a pointwise weak-majorization inequality for linear maps on Euclidean Jordan algebras and characterizes the least majorizing vector, extending classical matrix results.

Findings

Existence of the least vector in the weak-majorization set.

Least vector characterized as the join of eigenvalue vectors for positive maps.

Application to spectral norm estimates for linear maps.

Abstract

Given a linear map $T$ on a Euclidean Jordan algebra of rank $n$, we consider the set of all nonnegative vectors $q$ in $R^n$ with decreasing components that satisfy the pointwise weak-majorization inequality $\lambda(|T(x)|)\underset{w}{\prec}q*\lambda(|x|)$, where $\lambda$ is the eigenvalue map and $*$ denotes the componentwise product in $R^n$. 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 $T^*(e)$, 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 $n\times n$ 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.