Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the connectedness of spectral sets and irreducibility of spectral cones in Euclidean Jordan algebras
90
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let V be a Euclidean Jordan algebra of rank n. The eigenvalue map from V to R^n takes any element x in V to the vector of eigenvalues of x written in the decreasing order. A spectral set in V is the inverse image of a permutation set in R^n under the eigenvalue map. If the permutation set is also a convex cone, the spectral set is said to be a spectral cone. This paper deals with connectedness and arcwise connectedness properties of spectral sets. By relying on the result that in a simple Euclidean Jordan algebra, every eigenvalue orbit is arcwise connected, we show that if a permutation invariant set is connected (arcwise connected), then the corresponding spectral set is connected (respectively, arcwise connected). A related result is that in a simple Euclidean Jordan algebra, every pointed spectral cone is irreducible.
rate research
Read More
The commutation principle of Ramirez, Seeger, and Sossa proved in the setting of Euclidean Jordan algebras says that when the sum of a real valued function $h$ and a spectral function $Phi$ is minimized/maximized over a spectral set $E$, any local optimizer $a$ at which $h$ is Fr{e}chet differentiable operator commutes with the derivative $h^{prime}(a)$. In this paper, assuming the existence of a subgradient in place the derivative (of $h$), we establish `strong operator commutativity relations: If $a$ solves the problem $underset{E}{max},(h+Phi)$, then $a$ strongly operator commutes with every element in the subdifferential of $h$ at $a$; If $E$ and $h$ are convex and $a$ solves the problem $underset{E}{min},h$, then $a$ strongly operator commutes with the negative of some element in the subdifferential of $h$ at $a$. These results improve known (operator) commutativity relations for linear $h$ and for solutions of variational inequality problems. We establish these results via a geometric commutation principle that is valid not only in Euclidean Jordan algebras, but also in the broader setting of FTvN-systems.
Motivated by Horns 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 $lambda(sqrt{A}Bsqrt{A}) underset{log}{prec} lambda(A)*lambda(B)$ for positive semidefinite matrices and $lambda(|Acirc B|) underset{w}{prec} lambda(|A|)*lambda(|B|)$ for general (Hermitian) matrices, where $Acirc B$ 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 $lambdabig (P_{sqrt{a}}(b)big )underset{log}{prec} lambda(a)*lambda(b)$ for $a,bgeq 0$ and $lambdabig (|acirc b|big )underset{w}{prec} lambda(|a|)*lambda(|b|)$ 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 $lambda(|Abullet b|)underset{w}{prec} lambda({mathrm{diag}}(A))*lambda(|b|)$, 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 $||acirc b||_pleq ||a||_r,||b||_s$, where $||x||_p:=||lambda(x)||_p$ denotes the spectral $p$-norm of $x$ and $p,q,rin [1,infty]$ 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.
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 $ntimes n$ 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.
We show that for every pair of matrices (S,P), having the closed symmetrized bidisc $Gamma$ as a spectral set, there is a one dimensional complex algebraic variety $Lambda$ in $Gamma$ such that for every matrix valued polynomial f, the norm of f(S,P) is less then the sup norm of f on $Lambda$. The variety $Lambda$ is shown to have a particular determinantal representation, related to the so-called fundamental operator of the pair (S,P). When (S,P) is a strict $Gamma$-contraction, then $Lambda$ is a distinguished variety in the symmetrized bidisc, i.e., a one dimensional algebraic variety that exits the symmetrized bidisc through its distinguished boundary. We characterize all distinguished varieties of the symmetrized bidisc by a determinantal representation as above.
We show that a natural notion of irreducibility implies connectedness in the Compact Quantum Group setting. We also investigate the converse implication and show it is related to Kaplanskys conjectures on group algebras.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
