Dynamics truncated at total occupation

On $P_m({\mathcal{H}}_S\otimes {\mathcal{H}}_B)$, set

It is bounded and self-adjoint, since the reservoir energy is at most $m\Lambda$ there and $\Vert P_{m}V{P}_m\Vert \le 2G\sqrt{m+1}$. Let ${V_t}^{(m)}$ be its complete prepared isometry, embedded into the original output space.