arXiv subjects

Muddappa Gowda

Publications and source records attributed to Muddappa Gowda.

6 recordsLinked to original sources

arXiv2023

Transfer principles, Fenchel conjugate and subdifferential formulas in Fan-Theobald-von Neumann systems

A Fan-Theobald-von Neumann system is a triple $(V,W,λ)$, where $V$ and $W$ are real inner product spaces and $λ:V\to W$ is a norm-preserving map satisfying a Fan-Theobald-von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decomposition systems (Eaton triples). The present article is a continuation of an earlier paper, where the concepts of commutativity, automorphisms, majorization, and reduction were introduced and elaborated. Here, we describe some transfer principles and present Fenchel conjugate and subdifferential formulas.

math.FA↗

arXiv2022

Korovkin-type results and doubly stochastic transformations over Euclidean Jordan algebras

A well-known theorem of Korovkin asserts that if $\{T_k\}$ is a sequence of positive linear transformations on $C[a,b]$ such that $T_k(h)\rightarrow h$ (in the sup-norm on $C[a,b]$) for all $h\in \{1,ϕ,ϕ^2\}$, where $ϕ(t)=t$ on $[a,b]$, then $T_k(h)\rightarrow h$ for all $h\in C[a,b]$. In particular, if $T$ is a positive linear transformation on $C[a,b]$ such that $T(h)=h$ for all $h\in \{1,ϕ,ϕ^2\}$, then $T$ is the Identity transformation. In this paper, we present some analogs of these results over Euclidean Jordan algebras. We show that if $T$ is a positive linear transformation on a Euclidean Jordan algebra $V$ such that $T(h)=h$ for all $h\in \{e,p,p^2\}$, where $e$ is the unit element in $V$ and $p$ is an element of $V$ with distinct eigenvalues, then $T=T^*=I$ (the Identity transformation) on the span of the Jordan frame corresponding to the spectral decomposition of $p$; consequently, if a positive linear transformation coincides with the Identity transformation on a Jordan frame, then it is doubly stochastic. We also present sequential and weak-majorization versions.

math.FA↗

arXiv2020

Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras

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 $Φ$ is minimized/maximized over a spectral set $E$, any local optimizer $a$ at which $h$ is Fréchet 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 $\underset{E}{\max}\,(h+Φ)$, 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.

math.OC↗

arXiv2020

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 $λ(|T(x)|)\underset{w}{\prec}q*λ(|x|)$, where $λ$ 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.

math.FA↗

arXiv2020

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 $λ(\sqrt{A}B\sqrt{A}) \underset{log}{\prec} λ(A)*λ(B)$ for positive semidefinite matrices and $λ(|A\circ B|) \underset{w}{\prec} λ(|A|)*λ(|B|)$ 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 $λ\big (P_{\sqrt{a}}(b)\big )\underset{log}{\prec} λ(a)*λ(b)$ for $a,b\geq 0$ and $λ\big (|a\circ b|\big )\underset{w}{\prec} λ(|a|)*λ(|b|)$ for all $a$ and $b$, where $P_u$ and $λ(u)$ denote, respectively, the quadratic representation and the eigenvalue vector of an element $u$. We also describe inequalities of the form $λ(|A\bullet b|)\underset{w}{\prec} λ({\mathrm{diag}}(A))*λ(|b|)$, 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 Hölder type inequality $||a\circ b||_p\leq ||a||_r\,||b||_s$, where $||x||_p:=||λ(x)||_p$ 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.

math.FA↗

arXiv2018

On the connectedness of spectral sets and irreducibility of spectral cones in Euclidean Jordan algebras

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.

math.FA↗