A H\"older type inequality and an interpolation theorem in Euclidean Jordan algebras

In a Euclidean Jordan algebra V of rank n which carries the trace inner product, to each element x we associate the eigenvalue vector whose components are the eigenvalues of x written in the decreasing order. For any number p between (and including) one and infinity, we define the spectral p-norm of x to be the p-norm of the corresponding eigenvalue vector in the Euclidean n-space. In this paper, we show that for any two elements x and y, the one-norm of the Jordan product xoy is less than or equal to the product of p-norm of x and q-norm of y, where q is the conjugate of p. For a linear transformation on V, we state and prove an interpolation theorem relative to these spectral norms. In addition, we compute/estimate the norms of Lyapunov transformations, quadratic representations, and positive transformations on V

Some log and weak majorization inequalities in Euclidean Jordan algebras

Motivated by Horn's log-majorization (singular value) inequality $s(AB)\underset{log}{\prec} s(A)*s(B)$ and the related weak-majorization inequality $s(AB)\underset{w}{\prec} s(A)*s(B)$ for square complex matrices, we consider their Hermitian analogs $\lambda(\sqrt{A}B\sqrt{A}) \underset{log}{\prec} \lambda(A)*\lambda(B)$ for positive semidefinite matrices and $\lambda(|A\circ B|) \underset{w}{\prec} \lambda(|A|)*\lambda(|B|)$ for general (Hermitian) matrices, where $A\circ B$ denotes the Jordan product of $A$ and $B$ and $*$ denotes the componentwise product in $R^n$. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form $\lambda\big (P_{\sqrt{a}}(b)\big )\underset{log}{\prec} \lambda(a)*\lambda(b)$ for $a,b\geq 0$ and $\lambda\big (|a\circ b|\big )\underset{w}{\prec} \lambda(|a|)*\lambda(|b|)$ for all $a$ and $b$, where $P_u$ and $\lambda(u)$ denote, respectively, the quadratic representation and the eigenvalue vector of an element $u$. We also describe inequalities of the form $\lambda(|A\bullet b|)\underset{w}{\prec} \lambda({\mathrm{diag}}(A))*\lambda(|b|)$, where $A$ is a real symmetric positive semidefinite matrix and $A\,\bullet\, b$ is the Schur product of $A$ and $b$. In the form of an application, we prove the generalized H\"{o}lder type inequality $||a\circ b||_p\leq ||a||_r\,||b||_s$, where $||x||_p:=||\lambda(x)||_p$ denotes the spectral $p$-norm of $x$ and $p,q,r\in [1,\infty]$ with $\frac{1}{p}=\frac{1}{r}+\frac{1}{s}$. We also give precise values of the norms of the Lyapunov transformation $L_a$ and $P_a$ relative to two spectral $p$-norms.Comment: 18 pages; abstract, introduction, and final section rewritten, references update

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

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.Comment: 21 page