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Spectral Invariance and Submultiplicativity for the Algebras of S(M, g)-pseudo-differential Operators on Manifolds

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Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations

Part of the book series: Operator Theory: Advances and Applications ((APDE,volume 145))

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Abstract

For appropriate triples (M, g, M), where M is an (in general non-compact) manifold, g is a metric on T*M, and M is a weight function on T* M, we developed in [5] a pseudo-differential calculus on.A.4 which is based on the S(M, g)-calculus of L. Hörmander [30] in local models. Here we prove that the algebra of operators of order zero is a submultiplicative Ψ*-algebra in the sense of B. Gramsch [21] in \( \mathcal{L}\left( {{L^2}\left( M \right)} \right)\). For the basic calculus we generalized the concept of E. Schrohe [40] of so-called SG-compatible manifolds. In the proof of the existence of “order reducing operators” we apply a method from [4], and the proof of spectral invariance and submultiplicativity uses methods and results from B. Gramsch, J. Ueberberg, and K. Wagner [26] building on Beals’-type characterizations.

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Authors and Affiliations

  1. Fachbereich Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, Mainz, D-55099, Germany

    F. Baldus

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  1. F. Baldus
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Editors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Bonn, 53115, Germany

    Sergio Albeverio

  2. Institut für Mathematik, Technische Universität Clausthal, Clausthal-Zellerfeld, 38678, Germany

    Michael Demuth

  3. Institut für Mathematik, Universität Potsdam, Potsdam, 14415, Germany

    Elmar Schrohe  & Bert-Wolfgang Schulze  & 

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Baldus, F. (2003). Spectral Invariance and Submultiplicativity for the Algebras of S(M, g)-pseudo-differential Operators on Manifolds. In: Albeverio, S., Demuth, M., Schrohe, E., Schulze, BW. (eds) Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations. Operator Theory: Advances and Applications, vol 145. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8073-2_5

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  • DOI: https://doi.org/10.1007/978-3-0348-8073-2_5

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