A pseudo-differential calculus on non-standard symplectic space; spectral and regularity results in modulation spaces

The usual Weyl calculus is intimately associated with the choice of the standard symplectic structure on Rn⊕Rn. In this paper we will show that the replacement of this structure by an arbitrary symplectic structure leads to a pseudo-differential calculus of operators acting on functions or distributions defined, not on Rn but rather on Rn⊕Rn. These operators are intertwined with the standard Weyl pseudo-differential operators using an infinite family of partial isometries of L2(Rn)→L2(R2n) indexed by S(Rn). This allows us to obtain spectral and regularity results for our operators using Shubinʼs symbol classes and Feichtingerʼs modulation spaces.