Global normal forms and global properties in function spaces for second order Shubin type operators

We investigate the reduction to global normal forms of second order Shubin(or G) type differential operators P(x;D) in functional spaces on Rn. We describe the isomorphism properties of normal form transformations, introduced by L. Hormander for the study of affine symplectic transformations acting on pseudodifferential operators, in spaces like the Schwartz class, the weighted Shubin-Sobolev spaces and the Gelfand-Shilov spaces. We prove that the operator P(x;D) and the normal form PNF(x;D) have the same regularity/solvability and spectral properties. We also study the stability of global properties of the normal forms under perturbations by zero order Shubin type pseudodifferential operators and, more generally, by operators acting on S(Rn) and admitting discrete representations. Finally, we study Cauchy problems on Rn globally in time for second order hyperbolic equations P(x;D)+R(x;D), where P(x;D) is a second or der self-adjoint globally elliptic Shubin pseudodifferential operator and R(x;D) is a first order pseudodifferential operator.