Abstract
The pseudodifferential operators with symbols in the Grushin classes \~S supρ,δinf0 , 0 ≤ δ < ρ ≤ 1, of slowly varying symbols are shown to form spectrally invariant unital Frécher-*-algebras (Ψ*-algebras) in L(L 2(Rn)) and in L(H γ st) for weighted Sobolev spaces H supstinfγ defined via a weight d function γ. In all cases, the Fredholm property of an operator can be characterized by uniform ellipticity of the symbol. This gives a converse to theorems of Grushin and Kumano-Ta-Taniguchi. Both, the spectrum and the Fredholm spectrum of an operator turn out to be independent of the choices of s, t and γ.
The characterization of the Fredholm property by uniform ellipticity leads to an index theorem for the Fredholm operators in these classes, extending results of Fedosov and Hörmander.
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References
Atiyah, M.; Singer, I.: The Index of Elliptic Operators I. Ann. Math. 87 (1968), 484–539.
Beals, R.: Characterization of Pseudodifferential Operators and Applications. Duke Math. J. 44(1977), 45–77, 46(1979), 215.
Beals, R.: Weighted Distribution Spaces and Pseudodifferential Operators. Journal d'Analyse Math. 39 (1981), 131–187.
Bost, J.-B.: Principe d'Oka, K-théorie et Systémes Dynamiques Noncommutatifs. Invent. Math. 101 (1990), 261–333.
Connes, A.: C*-algébres et géometrie différentielle. C.R. Acad. Sc. Paris 290 ser. A (1980), 599–604.
Cordes, H. O.: On a Class of C*-Algebras. Math. Ann. 170 (1967), 283–313.
Cordes, H. O.: Elliptic Pseudo-Differential Operators — an Abstract Theory. LNM Springer 756, Berlin, Heidelberg New York 1979.
Cordes, H. O.: On Some C*-Algebras and Fréchet-*-Algebras of Pseudodifferential Operators. Proc. Sympos. Pure Math. 43 (1985), 79–104.
Cordes, H. O.: On Pseudodifferential Operators and Smoothness of Special Lie-Group Representations. Manuscripta Math.28 (1979), 51–69.
Cordes, H. O.: Spectral Theory of Linear Differential Operators and Comparison Algebras. London Math. Soc. Lecture Notes Series 76, Cambridge University Press, Cambridge, London, New York 1978.
Davies, E. B.; Simon, B.; Taylor, M.: Lρ Spectral Theory of Kleinian Groups. J. Funct. Anal. 78 (1988), 116–136.
Fedosov, B. V.: Analytical Formulas for the Index of Elliptic Operators. Trans. Moscow Math. Soc. 30 (1974), 159–240.
Gohberg, I.: On the Theory of Multidimensional Singular Integral Equations. Sov. Math. Dokl. 1 (1960), 960–963.
Gohberg, I.; Krupnik, N.: Einführung in die Theorie der eindimensionalen Integraloperatoren. Basel: Birkhäuser 1979.
Gramsch, B.: Relative Inversion in der Störungstheorie von Operatoren und Ψ-Algebren. Math. An. 269 (1984), 27–71.
Gramsch, B.; Kaballo, W.: Decompositions of Meromorphic Fredholm Resolvents and Ψ*-Algebras. Integral Equations Operator Theory 12 (1989), 23–41.
Gramsch, B.; Kalb, K.G.: Pseudo-Locality and Hypoellipticity in Operator Algebras. Semesterberichte Funktionalanalysis, Tübingen, Sommersemester 1985.
Gramsch, B.; Lorentz, K;; Scheiba, J.: Geodesics in Special Fréchet Manifolds. In preparation.
Gramsch, B.; Ueberberg, J.; Wagner, K.: Spectral Invariance and Submultiplicativity for Fréchet Algebras with Application to Algebras of Pseudo-Differential Operators. To appear in: Proceedings International Symposium Operator Calculus and Spectral Theory, Lambrecht, Dec. 1991.
Grushin, V. V.: Pseudo-differential Operators on fRn and Applications. Funct. Anal. Appl. 4 4 (1970), 202–212.
Hörmander, L.: Pseudodifferential Operators and Hypoelliptic Equations. Proc. Sympos. Pure Math. 10 (1967), 138–183.
Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Springer, Berlinn, New York, Tokyo 1985.
Kohn, J. J.; Nirenberg, L.: An Algebra of Pseudo-Differential Operators. Comm. Pure Appl. Math. 18 (1965), 269–305.
Kumano-go, H.: Pseudo-Differential Operators. MIT-Press, Cambridge, Mass., and London 1981.
Kumano-go, H.; Taniguchi, K.: Oscillatory If Symbols of Pseudodifferential Operators on Rn and Operators of Fredholm Type. Proc. Jap. Acad. 49 (1973), 397–402.
Leopold, H.-G.; Schrohe, E.: Spectral Invariance for Algebras of Pseudodifferential O Operators on Besov Spaces of Variable Order of Differentiation. To appear in math. Nachr.
Lockhart, R.; McOwen, R.: Elliptic Differential Operators on Noncompact Manifolds. Ann. Sc. Norm. Sup. Pisa, ser. IsnIV vol. XII (1985), 409–447.
McOwen, R.: On Elliptic Operators in Rn. Comm. Partial Differential Equations 5 (1980), 913–933.
Nirenberg, L.; Walker, H. F.: The Nullspaces of Elliptic Partial Differential Operators in Rn. J. Math Anal. Appl. 42 (1973), 271–301.
Schrohe, E.: Boundedness and Spectral Invariance for Standard Pseudodifferential Operators on Anisttropically Weighted Lρ-Sobolev Spaces. Integral Equations Operator Theory 13 (1990), 271–284.
Schrohe, E.: A Pseudodifferential Calculus for Weighted Symbols and a Fredholm Criterion for Boundary Value Problems on Noncompact Manifolds. Habilitationsschrift, Mainz 1991.
Schrohe, E.: The Symbols of an Algebra of Pseudodifferential Operators. Pac. Journal Math. 125 (1986), 211–224.
Schrohe, E.: A Ψ* Algebra of Pseudodifferential Operators on Noncompact manifolds. Arch. Math. 51 (1988), 81–86.
Schulze, B. W.: Topologies and Invertibility in Operator Spaces with Symbolic Structures. Proc. 9. TMP Karl-Marx-Stadt, Teubner Texte zur Mathematik, vol. 111, 1989.
Shubin, M. A.: Almost Periodic Functions and Partial Differential Operators. Russian Math. Surveys 33 (1978), 1–52.
Taylor, A. E.: Introduction to Functional Analysis. Wiley & Sons, New York 1967.
Taylor, M.: Gelfand Theory of Pseudo-Differential Operators and Hypoelliptic Operators. Trans. Amer. Math. Soc. 153 (1971), 495–510.
Uebergerg, J.: Zur Spektralinvarianz von Algebren von Pseudodifferentialoperatoren in der L ρ-Theorie. Manuscripta Math. 61 (1988), 459–475.
Widom, H.: Singular Integral Equations in L ρ. Trans. Amer. Math. Soc. 97 (1960), 131–160.
Wong, M. W.: Fredholm Pseudo-Differential Operators on Weighted Sobolev Spaces. Ark. Mat. 21 (1983), 271–282.
Wong, M. W.: Essential Spectra of Elliptic Pseudo-Differential Operators. Comm. Partial Differential Equations 13 (1988), 1209–1221.
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Communicated by B. W. Schulze
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Schrohe, E. Spectral invariance, ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces. Ann Glob Anal Geom 10, 237–254 (1992). https://doi.org/10.1007/BF00136867
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DOI: https://doi.org/10.1007/BF00136867