Lyapunov-like transformations on the tensor product of nuclear pairs of proper cones

Lyapunov-like transformation/matrix on a cone appears in the theory of dynamical systems and linear complementarity problems. The set of all Lyapunov-like transformations on a proper cone in a finite dimensional inner product space is the Lie algebra of the automorphism group of that cone. The dimension of this Lie algebra is called the Lyapunov rank. A pair of proper cones is said to be a nuclear pair if one of them is simplicial. In this paper, we find the Lyapunov rank and Lyapunov-like transformations on the tensor product of nuclear pairs of cones. Further, we prove that the space of Lyapunov-like transformations on the tensor product of a nuclear pair is the tensor product of the spaces of Lyapunov-like transformations on the individual cones. As a consequence, given a nuclear pair $(K_1,K_2)$, we describe the space of Lyapunov-like transformations on the cone of positive operators between $K_1$ and $K_2$.