On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras

The P, Z, and S properties of a linear transformation on a Euclidean Jordan algebra are generalizations of the corresponding properties of a square matrix on Rn. Motivated by the equivalence of P and S properties for a Z-matrix [2] and a similar result for Lyapunov and Stein transformations on the space of real symmetric matrices [6,5], in this paper, we present two results supporting the conjecture that P and S properties are equivalent for a Z-transformation on a Euclidean Jordan algebra. We show that the conjecture holds for Lyapunov-like transformations and Z-transformations satisfying an additional condition.