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

Author: Maurice A. de Gosson
Journal: Trans. Amer. Math. Soc. 365 (2013), 3287-3307
MSC (2010): Primary 47G30; Secondary 35Q40, 65P10, 35S05, 42B10
Published electronically: October 4, 2012
MathSciNet review: 3034466
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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 $ \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 $ \tau $-operators over the interval $ [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.

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Additional Information

Maurice A. de Gosson
Affiliation: Fakultät für Mathematik, Numerical Harmonic Analysis Group, Universität Wien, A-1090 Vienna, Austria

Keywords: Pseudo-differential operators, metaplectic group, symplectic covariance
Received by editor(s): May 6, 2011
Received by editor(s) in revised form: November 5, 2011
Published electronically: October 4, 2012
Dedicated: Dedicated to H.G. Feichtinger on his 60th birthday
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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