Generalized Weyl correspondence and Moyal multiplier algebras

We show that the Weyl correspondence and the concept of a Moyal multiplier can be naturally extended to generalized function classes that are larger than the class of tempered distributions. This generalization is motivated by possible applications to noncommutative quantum field theory. We prove that under reasonable restrictions on the test function space $E\subset L^2$, any operator in $L^2$ with a domain $E$ and continuous in the topologies of $E$ and $L^2$ has a Weyl symbol, which is defined as a generalized function on the Wigner–Moyal transform of the projective tensor square of $E$. We also give an exact characterization of the Weyl transforms of the Moyal multipliers for the Gel'fand–Shilov spaces $S^\beta_\beta$.