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.



Second quantization of partial isometry
For any contraction $X$ on $H$, the operator $\Gamma(X)$ (in Alain's notation S(X)) is a contraction. It is essentially algebraic to check that a pair of contractions $X,Y$ on $H$ will satisfy $\Gamma(X)\Gamma(Y)=\Gamma(XY)$ and $\Gamma(X)^\\*=\Gamma(X^\\*)$. So yes, you do get a partial isometry.


See Parthasarathy's book *An Introduction to Quantum Stochastic Calculus* (especially Exercise 20.22) for a very readable account of these facts.


(In fact, for any closed densely defined operator $X$ you can define a second quantisation $\Gamma(X)$ which is again closed and densely defined. It is bounded precisely when $X$ is a contraction.)


