No Arabic abstract
The commutation principle of Ramirez, Seeger, and Sossa cite{ramirez-seeger-sossa} proved in the setting of Euclidean Jordan algebras says that when the sum of a Fr{e}chet differentiable function $Theta(x)$ and a spectral function $F(x)$ is minimized over a spectral set $Omega$, any local minimizer $a$ operator commutes with the Fr{e}chet derivative $Theta^{prime}(a)$. In this paper, we extend this result to sets and functions which are (just) invariant under algebra automorphisms. We also consider a similar principle in the setting of normal decomposition systems.
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.
We study linear spaces of symmetric matrices whose reciprocal is also a linear space. These are Jordan algebras. We classify such algebras in low dimensions, and we study the associated Jordan loci in the Grassmannian.
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.
We classify, up to isomorphism, the 2-dimensional algebras over a field K. We focuse also on the case of characteristic 2, identifying the matrices of GL(2,F_2) with the elements of the symmetric group S_3. The classification is then given by the study of the orbits of this group on a 3-dimensional plane, viewed as a Fano plane. As applications, we establish classifications of Jordan algebras, algebras of Lie type or Hom-Associative algebras.
Let $T$ be a $delta$-Jordan Lie supertriple system. We first introduce the notions of generalized derivations and representations of $T$ and present some properties. Also, we study the low dimension cohomology and the coboundary operator of $T$, and then we investigate the deformations and Nijenhuis operators of $T$ by choosing some suitable cohomology.