Symplectic covariance properties for Shubin and Born–Jordan pseudo-differential operators

Abstract

Among all classes of pseudo-differential operators only the Weyl operators enjoy the property of symplectic covariance with respect to conjugation by elements of the metaplectic group. In this paper we show that there is, however, a weaker form of symplectic covariance for Shubin’s τ \tau -dependent operators, in which the intertwiners are no longer metaplectic, but are still invertible non-unitary operators. We also study the case of Born–Jordan operators, which are obtained by averaging the τ \tau -operators over the interval [ 0 , 1 ] [0,1] (such operators have recently been studied by Boggiatto and his collaborators, and by Toft). We show that covariance still holds for these operators with respect to a subgroup of the metaplectic group.