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The Invertible Double of Elliptic Operators

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Abstract

First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept of the Calderón projection. Then we summarize a recent construction of a canonical invertible double for general first order elliptic differential operators over smooth compact manifolds with boundary. We derive a natural formula for the Calderón projection which yields a generalization of the famous Cobordism Theorem. We provide a list of assumptions to obtain a continuous variation of the Calderón projection under smooth variation of the coefficients. That yields various new spectral flow theorems. Finally, we sketch a research program for confining, respectively closing, the last remaining gaps between the geometric Dirac operator type situation and the general linear elliptic case.

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Author notes

  1. Matthias Lesch

    Present address: Department of Mathematics, University of Colorado at Boulder, Boulder, CO, 80309-0395, USA

Authors and Affiliations

  1. Department of Science, Systems, and Models, Roskilde University, 4000, Roskilde, Denmark

    Bernhelm Booss-Bavnbek

  2. Mathematisches Institut, Universität Bonn, Beringstr. 6, 53115, Bonn, Germany

    Matthias Lesch

Authors
  1. Bernhelm Booss-Bavnbek
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  2. Matthias Lesch
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Corresponding author

Correspondence to Bernhelm Booss-Bavnbek.

Additional information

This short review and work program is dedicated to the memory of Krzysztof P. Wojciechowski (1953–2008), who was a leader of the investigation of spectral invariants of Dirac type operators for almost 30 years.

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Booss-Bavnbek, B., Lesch, M. The Invertible Double of Elliptic Operators. Lett Math Phys 87, 19–46 (2009). https://doi.org/10.1007/s11005-009-0292-9

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  • DOI: https://doi.org/10.1007/s11005-009-0292-9

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