Semigroups for quadratic evolution equations acting on Shubin–Sobolev and Gelfand–Shilov spaces

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