Characterization of p-property for some z-transformations on positive semidefinite cone

The P-property of the following two Z-transformations with respect to the positive semidefinite cone is characterized: (i) I-S, where S: Sn×n→Sn×n is a nilpotent linear transformation, (ii) I-L-1 A, where LA is the Lyapunov transformation defined on Sn×n by LA(X) = AX + XAT. (Here Sn×n denotes the space of all symmetric n×n matrices and I is the identity transformation.).