On linear maps leaving invariant the copositive/completely positive cones

The objective of this manuscript is to investigate the structure of linear maps on the space of real symmetric matrices S n that leave invariant the closed convex cones of copositive and completely positive matrices (COPn and CPn). A description of an invertible linear map on S n such that L(CPn). CPn is obtained in terms of semipositive maps over the positive semidefinite cone S n + and the cone of symmetric nonnegative matrices N n + for n 6 4, with specific calculations for n = 2. Preserver properties of the Lyapunov map X 7. AX + XAt, the generalized Lyapunov map X 7. AXB + BtXAt, and the structure of the dual of the cone (CPn) (for n 6 4) are brought out. We also highlight a different way to determine the structure of an invertible linear map on S2 that leaves invariant the closed convex cone S2 +.