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 [Formula: see text]. 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 [Formula: see text] but rather on [Formula: see text]. These operators are intertwined with the standard Weyl pseudo-differential operators using an infinite family of partial isometries of [Formula: see text] indexed by [Formula: see text]. This allows us to obtain spectral and regularity results for our operators using Shubinʼs symbol classes and Feichtingerʼs modulation spaces.