Characterization of the Moyal Multiplier Algebras for the Generalized Spaces of Type $S$

We investigate the properties of the Moyal multiplier algebras for the generalized Gelfand–Shilov spaces $S^{b_n}_{a_k}$. We prove that these algebras contain Palamodov spaces of type $\mathscr E$, and establish continuity properties of the operators with Weyl symbols in this class. Analogous results are obtained for the projective version of the spaces of type $S$ and are extended to the multiplier algebras for various translation-invariant star products.