Almost Diagonalization of Pseudodifferential Operators

Abstract

In this review we focus on the almost diagonalization of pseudodifferential operators and highlight the advantages that time-frequency techniques provide here. In particular, we retrace the steps of an insightful paper by Gröchenig, who succeeded in characterizing a class of symbols previously investigated by Seöstrand by noticing that Gabor frames almost diagonalize the corresponding Weyl operators. This approach also allows to give new and more natural proofs of related results such as boundedness of operators or algebra and Wiener properties of the symbol class. Then, we discuss some recent developments on the theme, namely an extension of these results to a more general family of pseudodifferential operators and similar outcomes for a symbol class closely related to Sjöstrand's one. This is a precopyedited version of a contribution published in Landscapes ofTime-Frequency Analysis (Boggiatto P. et al. (eds)) published by Birkhäuser, Cham. The definitive authenticated version is available online via https://doi.org/10.1007/978-3-030-05210-2 14.