Banach algebras of pseudodifferential operators and their almost diagonalization

Gröchenig, Karlheinz; Rzeszotnik, Ziemowit

doi:10.5802/aif.2414

Banach algebras of pseudodifferential operators and their almost diagonalization
[Algèbres de Banach d’opérateurs pseudo-différentiels et leur presque diagonalisation]

Gröchenig, Karlheinz ¹ ; Rzeszotnik, Ziemowit ²

¹ University of Vienna Faculty of Mathematics Nordbergstrasse 15 1090 Wien (Austria)
² University of Wroclaw Mathematical Institute Pl. Grunwaldzki 2/4 50-384 Wroclaw (Poland)

Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 2279-2314.

Résumé
Abstract

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 $M^{\infty, 𝒜}$ . Chaque opérateur pseudo-différentiel dans $M^{\infty, 𝒜}$ 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 $M^{\infty, 𝒜}$ sont bornés sur $L^{2} (ℝ^{d})$ 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 $S_{0, 0}^{0}$ .

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 $M^{\infty, 𝒜}$ . Then every operator with a symbol in $M^{\infty, 𝒜}$ 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 $L^{2} (ℝ^{d})$ . 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 $S_{0, 0}^{0}$ .

MR Zbl

DOI : 10.5802/aif.2414

Classification : 42C40, 35S05
Keywords: Pseudodifferential operators, symbol class, symbolic calculus, Banach algebra, inverse-closedness, Wiener’s Lemma
Mot clés : opérateur pseudodifferentiel, classe de symboles, calcul symbolique, algèbre de Banach, lemme de Wiener

Affiliations des auteurs :

Gröchenig, Karlheinz ¹ ; Rzeszotnik, Ziemowit ²

¹ University of Vienna Faculty of Mathematics Nordbergstrasse 15 1090 Wien (Austria)
² University of Wroclaw Mathematical Institute Pl. Grunwaldzki 2/4 50-384 Wroclaw (Poland)

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     title = {Banach algebras of pseudodifferential operators and their almost diagonalization},
     journal = {Annales de l'Institut Fourier},
     pages = {2279--2314},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Gröchenig, Karlheinz; Rzeszotnik, Ziemowit. Banach algebras of pseudodifferential operators and their almost diagonalization. Annales de l'Institut Fourier, Tome 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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