Positive principal minor property of linear transformations on Euclidean Jordan algebras (Q1024238)

A Euclidean Jordan algebra is a finite dimension space \(V\) over \(\mathbb{R}\) with inner product \(\left\langle ~\right\rangle \) together with a commutative bilinear binary operation \(\circ\) with unity \(e\) such that \(x\circ(x^{2}\circ y)=x^{2}\circ(x\circ y)\) and \(\left\langle x\circ y,z\right\rangle =\left\langle y,x\circ z\right\rangle \) for all \(x,y,z\in V\). Every Euclidean Jordan algebra is a direct sum of simple Euclidean Jordan algebras, and the latter are completely classified in [\textit{J. Faraut}, and \textit{A. Korányi}, Analysis on symmetric cones. Oxford: Clarendon Press (1994; Zbl 0841.43002)]. The author considers various properties of linear operators on a Euclidean Jordan algebra \(V\), and in particular the property that all principal minors of the operator are positive. A typical result is the following (see Theorem 4.4). Put \(K:=\left\{ x\circ x\mid x\in V\right\} \). Then an automorphism \(L\) of a Euclidean Jordan space \(V\) has all of its principal minors positive if and only if for each \(q\in V\) there exists \(x\in K\) such that \(Lx+q\in K\) and \(\left\langle x,Lx+q\right\rangle =0\).