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 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 a general and efficient way to describe the algebraic and continuity properties of pseudodifferential operators with symbols having an exponential or super-exponential growth at infinity.