Banach algebras of pseudodifferential operators and their almost diagonalization
Annales de l'Institut Fourier, Volume 58 (2008) no. 7, pp. 2279-2314.

We define new symbol classes for pseudodifferential operators and investigate their pseudodifferential calculus. The symbol classes are parametrized by commutative convolution algebras. To every solid convolution algebra 𝒜 over a lattice Λ we associate a symbol class M ,𝒜 . Then every operator with a symbol in M ,𝒜 is almost diagonal with respect to special wave packets (coherent states or Gabor frames), and the rate of almost diagonalization is described precisely by the underlying convolution algebra 𝒜. Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded operators on L 2 ( d ). If a version of Wiener’s lemma holds for 𝒜, then the algebra of pseudodifferential operators is closed under inversion. The theory contains as a special case the fundamental results about Sjöstrand’s class and yields a new proof of a theorem of Beals about the Hörmander class S 0,0 0 .

Nous étudions une nouvelle classe de symboles pour les opérateurs pseudo-différentiels et leurs calculs symboliques. À chaque algèbre 𝒜 commutative par rapport aux convolutions sur un réseau Λ correspond une classe de symboles M ,𝒜 . Chaque opérateur pseudo-différentiel dans M ,𝒜 est presque diagonale par rapport aux états cohérents, et sa décroissance hors de la diagonale est décrite par l’algèbre 𝒜. Les opérateurs pseudo-différentiels avec des symboles dans M ,𝒜 sont bornés sur L 2 ( d ) et constituent une algèbre de Banach. Si une version du lemme de Wiener s’applique à 𝒜, alors l’algèbre d’opérateurs pseudo-différentiels est fermée par rapport à l’inversion des opérateurs. La théorie contient comme un cas spécial la théorie de J. Sjöstrand et fournit une nouvelle démonstration d’un théorème de Beals sur les symboles de Hörmander dans S 0,0 0 .

Received:
Accepted:
DOI: 10.5802/aif.2414
Classification: 42C40,  35S05
Keywords: Pseudodifferential operators, symbol class, symbolic calculus, Banach algebra, inverse-closedness, Wiener’s Lemma
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Gröchenig, Karlheinz; Rzeszotnik, Ziemowit. Banach algebras of pseudodifferential operators and their almost diagonalization. Annales de l'Institut Fourier, Volume 58 (2008) no. 7, pp. 2279-2314. doi : 10.5802/aif.2414. https://aif.centre-mersenne.org/articles/10.5802/aif.2414/

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