Symplectic covariance properties for Shubin and Born-Jordan pseudo-differential operators (Q2838123)

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. The principal aim of this paper is to report on the fact that there exists a weaker form of symplectic covariance for \(\tau\)-operators which reduces to the case above when \(\tau=\frac{1}{2}\). The authors show that there is a weaker form of symplectic covariance for Shubin's \(\tau\)-dependent operators, in which the intertwiners no longer are metaplectic, but still are invertible non-unitary operators. They also study the case of Born-Jordan operators, which are obtained by averaging the \(\tau\)-operators over the interval \([0,1]\). Such operators have recently been studied by \textit{P. Boggiatto} et al.\ [Trans. Am. Math. Soc. 362, No. 9, 4955--4981 (2010; Zbl 1200.47064); J. Pseudo-Differ. Oper. Appl. 1, No.~4, 401--415 (2010; Zbl 1214.47041); Bull. Lond. Math. Soc. 45, No.~6, 1131--1147 (2013; Zbl 1287.42007)]. The authors show that metaplectic covariance still hold for these operators, with respect top a subgroup of the metaplectic group.