Some inertia theorems in Euclidean Jordan algebras

This paper deals with some inertia theorems in Euclidean Jordan algebras. First, based on the continuity of eigenvalues, we give an alternate proof of Kaneyuki’s generalization of Sylvester’s law of inertia in simple Euclidean Jordan algebras. As a consequence, we show that the cone spectrum of any quadratic representation with respect to a symmetric cone is finite. Second, we present Ostrowski–Schneider type inertia results in Euclidean Jordan algebras. In particular, we relate the inertias of objects a and x in a Euclidean Jordan algebra when La(x)>0 or Sa(x)>0, where La and Sa denote Lyapunov and Stein transformations, respectively.