Hidden commutativity in semi-FTvN systems

Abstract

A semi-FTvN system (short for semi-Fan-Theobald-von Neumann system) is a triple $(\mathcal{V},\mathcal{W},\lambda)$, where $\mathcal{V}$ and $\mathcal{W}$ are real inner product spaces and the mapping $\lambda:\mathcal{V}\rightarrow \mathcal{W}$ satisfies the sharpened Cauchy-Schwarz inequality $\langle x,y\rangle\leq\langle \lambda(x),\lambda(y)\rangle \leq ||x||\,||y||$. Such a system arises, for example, from a complete hyperbolic system and is a generalization of Fan-Theobald-von Neumann system. In this article, we introduce two commutativity concepts: Strong commutativity via the equality $\langle x,y\rangle=\langle \lambda(x),\lambda(y)\rangle$, and commutativity via the condition $\langle Dx,y\rangle=0$ for all $D$ in the Lie algebra of the automorphism group of the system. We show that strong commutativity implies commutativity, and in the special case of an Euclidean Jordan algebra, commutativity and strong commutativity concepts reduce, respectively, to those of operator and strong operator commutativity. In the form of an application to optimization problems, we describe a commutation principle, where commutativity appears as an optimality condition.