It can be seen directly in the Fock model that the implementing operators can be chosen so that they define an ordinary representation of the double cover of H constructed above. The corresponding semigroup of contraction operators is called the oscillator semigroup. The extended oscillator semigroup is obtained by taking the semidirect product with the operators W (u). These operators lie in every Schatten class and leave invariant the Sobolev spaces and the space of entire vectors for W.