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.

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.)