  We provide two extensions of a dense subspace of Fock space, such that
Bogoliubov transformations become implementable on them, even though they
violate the Shale-Stinespring condition, so they are not implementable on Fock
space. Both the bosonic and fermionic case are covered. Conditions for
implementability in the extended sense are stated and proved. From these, we
derive conditions for a quadratic Hamiltonian to be diagonalizable by a
Bogoliubov transformation that is implementable in the extended sense. Three
examples illustrate situations, in which an implementation in the extended
sense is possible although the Shale-Stinespring condition fails to hold.
