Inclusion theorems for the Moyal multiplier algebras of generalized Gelfand-Shilov spaces

We prove that the Moyal multiplier algebras of the generalized Gelfand-Shilov spaces of type $S$ contain Palamodov spaces of type $\mathcal{E}$ and the inclusion maps are continuous. We also give a direct proof that the Palamodov spaces are algebraically and topologically isomorphic to the strong duals of the spaces of convolutors for the corresponding spaces of type $S$. The obtained results provide an effective way to describe the properties of pseudodifferential operators with symbols in the spaces of type $\mathcal{E}$.