Symplectic Covariance Properties for Shubin and Born-Jordan Pseudo-Differential Operators

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-dependent operators, in which the intertwiners no longer are metaplectic, but still are invertible non-unitary operators. We also study the case of Born--Jordan operators, which are obtained by averaging the τ-operators over the interval [0,1] (such operators have recently been studied by Boggiatto and his collaborators). We show that metaplectic covariance still hold for these operators, with respect top a subgroup of the metaplectic group.

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