  We consider the initial value Cauchy problem for a class of evolution
equations whose Hamiltonian is the Weyl quantization of a homogeneous quadratic
form with non-negative definite real part. The solution semigroup is shown to
be strongly continuous on several spaces: the Shubin--Sobolev spaces, the
Schwartz space, the tempered distributions, the equal index Beurling type
Gelfand--Shilov spaces and their dual ultradistribution spaces.
