Dissipative Hamiltonian representations of linear evolution equations

We investigate explicit and implicit dissipative Hamiltonian representations of generators of strongly continuous semigroups on real or complex Hilbert spaces. For a prescribed positive definite selfadjoint energy operator H satisfying dom(A)(H), we show that the Lyapunov dissipation inequality is equivalent to an exact factorization $A=DH$, where D can be chosen maximally dissipative. For bounded H, the criterion also yields an implicit representation. We further study energy constructions based on closed intertwiners and output trajectories, distinguishing contractive energy-space dynamics from operator factorizations on the original state space. Examples illustrate the roles of noncoercivity, unboundedness, and domain compatibility.