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Composition and spectral invariance of pseudodifferential Operators on Modulation Spaces

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Abstract

We introduce new classes of Banach algebras of pseudodifferential operators with symbols in certain modulation spaces and investigate their composition and the functional calculus. Operators in these algebras possess the spectral invariance property on the associated family of modulation spaces. These results extend and contain Sjöstrand's theory, and they are obtained with new phase-space methods instead of “hard analysis”.

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References

  1. R. Beals,Characterization of pseudodifferential operators and applications, Duke Math. J.44 (1977), 45–57.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Boggiatto and E. Schrohe,Characterization, spectral invariance and the Fredholm property of multi-quasi-elliptic operators, Rend. Sem. Mat. Univ. Politec. Torino59 (2003), 229–242.

    MathSciNet  Google Scholar 

  3. J.-M. Bony and J.-Y. Chemin,Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France122 (1994), 77–118.

    MATH  MathSciNet  Google Scholar 

  4. A. Boulkhemair,Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett.4 (1997), 53–67.

    MATH  MathSciNet  Google Scholar 

  5. E. Cordero and K. Gröchenig,Time-frequency analysis of localization operators, J. Funct. Anal.205 (2003), 107–131.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Cordero and K. Gröchenig,Necessary conditions for Schatten class localization operators, Proc. Amer. Math. Soc.133 (2005), 3573–3579.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Dimassi and J. Sjöstrand,Spectral Asymptotics in the Semi-Classical Limit, Cambridge University Press, Cambridge, 1999.

    MATH  Google Scholar 

  8. H. G. Feichtinger,Modulation spaces on locally compact abelian groups, inProceedings of “International Conference on Wavelets and Applications” 2002, Chennai, India, 2003, pp. 99–140. Updated version of a technical report, University of Vienna, 1983.

  9. H. G. Feichtinger and K. Gröchenig,Gabor frames and time-frequency analysis of distributions, J. Funct. Anal.146 (1997), 464–495.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Fendler, K. Gröchenig, M. Leinert, J. Ludwig and C. Molitor-Braun,Weighted group algebras on groups of polynomial growth, Math. Z.102 (2003), 791–821.

    Article  Google Scholar 

  11. G. Fendler, K. Gröchenig and M. Leinert,Symmetry of weighted L 1 -algebras and the GRS-condition, Bull. London Math. Soc., to appear.

  12. G. B. Folland,Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ, 1989.

    MATH  Google Scholar 

  13. I. Gel'fand, D. Raikov and G. Shilov,Commutative Normed Rings, Chelsea Publ. Co., New York, 1964.

    Google Scholar 

  14. B. Gramsch,Relative Inversion in der Störungstheorie von Operatoren und ϕ-Algebren, Math. Ann.269 (1984), 27–71.

    Article  MATH  MathSciNet  Google Scholar 

  15. K. Gröchenig,Foundations of Time-Frequency Analysis, Birkhäuser Boston Inc., Boston, MA, 2001.

    MATH  Google Scholar 

  16. K. Gröchenig,Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl.10 (2004), 105–132.

    Article  MATH  MathSciNet  Google Scholar 

  17. K. Gröchenig and C. Heil,Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory34 (1999), 439–457.

    Article  MathSciNet  Google Scholar 

  18. K. Gröchenig and C. Heil,Modulation spaces as symbol classes for pseudodifferential operators, inWavelets and Their Applications, (S. T. M. Krishna and R. Radha, eds.), Allied Publishers, Chennai, 2003, pp. 151–170.

    Google Scholar 

  19. K. Gröchenig and C. Heil,Counterexamples for boundedness of pseudodifferential operators, Osaka J. Math.41 (2004), 1–11.

    MathSciNet  Google Scholar 

  20. K. Gröchenig and M. Leinert,Wiener's lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc.17 (2004), 1–18.

    Article  MATH  MathSciNet  Google Scholar 

  21. C. Heil, J. Ramanathan and P. Topiwala,Singular values of compact pseudodifferential operators, J. Funct. Anal.150 (1997), 426–452.

    Article  MATH  MathSciNet  Google Scholar 

  22. F. Hérau,Melin-Hörmander inequality in a Wiener type pseudo-differential algebra, Ark. Mat.39 (2001), 311–338.

    Article  MATH  MathSciNet  Google Scholar 

  23. L. Hörmander,The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math.32 (1979), 360–444.

    Article  MathSciNet  Google Scholar 

  24. L. Hörmander,The Analysis of Linear Partial Differential Operators. III, Springer-Verlag, Berlin, 1985. Pseudodifferential operators.

    MATH  Google Scholar 

  25. A. Hulanicki,On the spectrum of convolution operators on groups with polynomial growth, Invent. Math.17 (1972), 135–142.

    Article  MATH  MathSciNet  Google Scholar 

  26. S. Jaffard,Propriétés des matrices “bien localisées” près de leur diagonalé et quelques applications, Ann. Inst. H. Poincaré Anal. Non Linéaire7 (1990), 461–476.

    MATH  MathSciNet  Google Scholar 

  27. D. Labate,Pseudodifferential operators on modulation spaces, J. Math. Anal. Appl.262 (2001), 242–255.

    Article  MATH  MathSciNet  Google Scholar 

  28. H.-G. Leopold and E. Schrohe,Spectral invariance for algebras of pseudodifferential operators on Besov-Triebel-Lizorkin spaces, Manuscripta Math.78 (1993), 99–110.

    Article  MATH  MathSciNet  Google Scholar 

  29. J. Sjöstrand,An algebra of pseudodifferential operators, Math. Res. Lett.1 (1994), 185–192.

    MATH  MathSciNet  Google Scholar 

  30. J. Sjöstrand,Wiener type algebras of pseudodifferential operators, inSéminaire sur les Équations aux Dérivées Partielles, 1994–1995, École Polytech., Palaiseau, 1995, Exp. No. IV, 21.

    Google Scholar 

  31. E. M. Stein,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.

    MATH  Google Scholar 

  32. T. Strohmer,On the role of the Heisenberg group in wireless communication, technical report, 2004.

  33. J. Toft,Subalgebras to a Wiener type algebra of pseudo-differential operators, Ann. Inst. Fourier (Grenoble)51 (2001), 1347–1383.

    MATH  MathSciNet  Google Scholar 

  34. J. Toft,Continuity properties in non-commutative convolution algebras, with applications in pseudo-differential calculus, Bull. Sci. Math.126 (2002), 115–142.

    Article  MATH  MathSciNet  Google Scholar 

  35. J. Ueberberg,Zur Spektralinvarianz von Algebren von Pseudodifferentialoperatoren in der L p -Theorie, Manuscripta Math.61 (1988), 459–475.

    Article  MATH  MathSciNet  Google Scholar 

  36. H. Weyl,The Theory of Groups and Quantum Mechanics, Methuen, London, 1931; reprinted by Dover Puplications, New York, 1950.

    MATH  Google Scholar 

  37. L. Zadeh,The determination of the impulsive response of variable networks, J. Appl. Phys.21 (1950), 642–645.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Karlheinz Gröchenig

    Present address: Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090, Wien, Austria

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  1. Institute of Biomathematics and Biometry, GSF-National Research Center for Environment and Health, Ingolstädter Landstrasse 1, 85764, Neuherberg, Germany

    Karlheinz Gröchenig

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Correspondence to Karlheinz Gröchenig.

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Gröchenig, K. Composition and spectral invariance of pseudodifferential Operators on Modulation Spaces. J. Anal. Math. 98, 65–82 (2006). https://doi.org/10.1007/BF02790270

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  • DOI: https://doi.org/10.1007/BF02790270

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