The Q-property of composite transformations and the P-property of Stein-type transformations on self-dual and symmetric cones

Motivated by the Q-property of nonsingular M-matrices, Lyapunov and Stein transformations (corresponding to positive stable and Schur stable matrices, respectively) and their products [A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994; M.S. Gowda, Y. Song, Math. Prog. 88 (2000) 575–587; M.S. Gowda, T. Parthasarathy, Linear Algebra Appl. 320 (2000) 131–144; M.S. Gowda, Y. Song, SIAM J. Matrix Anal. Appl. 24 (2002) 25–39], in the first part of the paper we present a unifying result on the product of Q-transformations defined on self-dual closed convex cones. The second part deals with the P-property of the linear transformation I − S on a Euclidean Jordan algebra where S leaves the corresponding symmetric cone invariant and ρ(S) < 1. We prove the P-property for the Lorentz cone and present some partial results in the general case.