ORDER PRESERVING AUTOMORPHISMS OF SHUBIN TYPE PSEUDODIFFERENTIAL OPERATORS

We consider the algebra of all classical Shubin type pseudodifferential operators on Euclidean space. Following up on a remarkable result of Duistermaat and Singer (Comm. Pure Appl. Math. XXVIV:39-47, 1976) for the case of the usual pseudodifferential operators, we determine its order preserving automorphisms. We show that they are given by conjugation with Fourier integral operators of a particular class. At first glance, this looks very much like the statement of Duistermaat  and Singer. There are, however, significant analytic and geometric differences. Joint work in progress with Robert Hesse (Hannover) and Ryszard Nest (Copenhagen).