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Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Autorschaft

  • Anton Savin
  • Elmar Schrohe

Organisationseinheiten

Externe Organisationen

  • Peoples' Friendship University of Russia (RUDN)

Details

OriginalspracheEnglisch
Aufsatznummer108400
FachzeitschriftJournal of functional analysis
Jahrgang278
Ausgabenummer5
PublikationsstatusVeröffentlicht - 15 März 2020

Abstract

We consider elliptic operators associated with discrete groups of quantized canonical transformations. In order to be able to apply results from algebraic index theory, we define the localized algebraic index of the complete symbol of an elliptic operator. With the help of a calculus of semiclassical quantized canonical transformations, a version of Egorov's theorem and a theorem on trace asymptotics for semiclassical Fourier integral operators we show that the localized analytic index and the localized algebraic index coincide. As a corollary, we express the Fredholm index in terms of the algebraic index for a wide class of groups, in particular, for finite extensions of Abelian groups.

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Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations. / Savin, Anton; Schrohe, Elmar.
in: Journal of functional analysis, Jahrgang 278, Nr. 5, 108400, 15.03.2020.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Savin A, Schrohe E. Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations. Journal of functional analysis. 2020 Mär 15;278(5):108400. doi: 10.48550/arXiv.1812.11550, 10.1016/j.jfa.2019.108400
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AU - Savin, Anton

AU - Schrohe, Elmar

N1 - Funding Information: The authors are grateful to M. Doll, A. Gorokhovsky, V. Nazaikinskii, R. Nest, T. Schick, and R. Schulz for useful discussions. This work was partially supported by Deutsche Forschungsgemeinschaft, grant SCHR 319/8-1, RFBR, grants 16-01-00373, 19-01-00574, and RUDN University program 5-100.

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N2 - We consider elliptic operators associated with discrete groups of quantized canonical transformations. In order to be able to apply results from algebraic index theory, we define the localized algebraic index of the complete symbol of an elliptic operator. With the help of a calculus of semiclassical quantized canonical transformations, a version of Egorov's theorem and a theorem on trace asymptotics for semiclassical Fourier integral operators we show that the localized analytic index and the localized algebraic index coincide. As a corollary, we express the Fredholm index in terms of the algebraic index for a wide class of groups, in particular, for finite extensions of Abelian groups.

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