If we have a unitary map from Hilbert space $H$ to $H$, we get a unitary map from $e^{H}$ to $e^{H}$, where $e^{H}$ is the symmetric Fock space of $H$. But if we replace the unitary with partial isometry, will we get partial isometry at the Fock space level?

My main aim is to study $E_0$ semigroup (on type I factor) coming from semigroup of partial isometries on some Hilbert space.