Hamiltonian actions on the cone of positive definite matrices

The semigroup of Hamiltonians acting on the cone of positive definite matrices via linear fractional transformations satisfies the Birkhoff contraction formula for the Thompson metric. In this paper we describe the action of the Hamiltonians lying in the boundary of the semigroup. This involves in particular a construction of linear transformations leaving invariant the cone of positive definite matrices (strictly positive linear mappings) parameterized over all square matrices. Its invertibility and relation to the Lyapunov and Stein operators are investigated in detail. In particular, it is shown that each of these linear transformations commutes with the corresponding Lyapunov operator and contracts the Thompson metric.