Bogoliubov Transformations Beyond Shale–Stinespring: Generic v∗v for Bosons

We construct an extension of Fock space and prove that it allows for implementing bosonic Bogoliubov transformations in a certain extended sense. While an implementation in the regular sense on Fock space is only possible if a certain operator $ v^*v $ is trace class (this is the well-known Shale-Stinespring condition), the extended implementation works without any restrictions on this operator. This generalizes a recent result of extended implementability, which required $ v^*v $ to have discrete spectrum.