Pseudodifferential parametrices of infinite order and SG-hyperbolic problems

In this paper we consider a class of symbols of infinite order and develop a global calculus for the related pseudodifferential operators in the functional frame of the Gelfand-Shilov spaces of type S. As an application, we construct a parametrix for the Cauchy problem associated to an operator with principal part $D_t^m$ and lower order terms given by SG-operators, cf. Introduction. We do not assume here Levi conditions on the lower order terms. Giving initial data in Gelfand-Shilov spaces, we are able to prove the well-posedness for the problem and to give an explicit expression of the solution.