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 $\operatorname{dom}(A)\subset\operatorname{dom}(H)$, 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.