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Russian Journal of Mathematical Physics

, Volume 22, Issue 3, pp 410–420 | Cite as

Uniformization and index of elliptic operators associated with diffeomorphisms of a manifold

  • A. Savin
  • E. Schrohe
  • B. Sternin
Article

Abstract

We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely, differential operators with shifts induced by the action of a (not necessarily periodic) isometric diffeomorphism. The key to the solution is the method of uniformization. To the nonlocal problem we assign a pseudodifferential operator, with the same index, acting on the sections of an infinite-dimensional vector bundle on a compact manifold. We then determine the index in terms of topological invariants of the symbol, using the Atiyah—Singer index theorem.

Keywords

Vector Bundle Elliptic Operator Fredholm Operator Noncommutative Geometry Topological Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. Antonevich and A. Lebedev, Functional-Differential Equations. I. C*-Theory, Number 70 in Pitman Monographs and Surveys in Pure and Applied Mathematics (Longman, Harlow, 1994).Google Scholar
  2. 2.
    A. Antonevich, M. Belousov and A. Lebedev, Functional-Differential Equations. II. C*-Applications, Number 94 in Pitman Monographs and Surveys in Pure and Applied Mathematics (Longman, Harlow, 1998).Google Scholar
  3. 3.
    M. F. Atiyah and I. M. Singer, “The Index of Elliptic Operators I,” Ann. of Math. 87, 484–530 (1968).MathSciNetCrossRefGoogle Scholar
  4. 4.
    P. Baum and A. Connes, “Chern Character for Discrete Groups,” in A Fête of Topology (Academic Press, Boston, MA, 1988). pp. 163–232.Google Scholar
  5. 5.
    A. Connes, Noncommutative Geometry (Academic Press, San Diego, CA, 1994).zbMATHGoogle Scholar
  6. 6.
    B. V. Fedosov, “Index Theorems,” Partial Differential Equations VIII, in Encyclopaedia Math. Sci. 65 (Springer, Berlin, 1996). pp. 155–251.MathSciNetGoogle Scholar
  7. 7.
    G. Kasparov, “Equivariant KK-Theory and the Novikov Conjecture,” Invent. Math. 91 (1), 147–201 (1988).MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    L. Hörmander, The Analysis of Linear Partial Differential Operators. III (Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1985).zbMATHGoogle Scholar
  9. 9.
    G. Luke, “Pseudodifferential Operators on Hilbert Bundles,” J. Diff. Equations 12, 566–589 (1972).zbMATHMathSciNetCrossRefADSGoogle Scholar
  10. 10.
    V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin, Elliptic Theory and Noncommutative Geometry (Birkhäuser Verlag, Basel, 2008).zbMATHGoogle Scholar
  11. 11.
    V. E. Nazaikinskii, A. Savin, B.-W. Schulze, and B. Sternin, Elliptic Theory on Singular Manifolds (CRC-Press, Boca Raton, 2005).zbMATHCrossRefGoogle Scholar
  12. 12.
    G. Rozenblum, “On Some Analytical Index Formulas Related to Operator-Valued Symbols,” Electron. J. Differential Equations (17), 1–31 (2002).zbMATHMathSciNetGoogle Scholar
  13. 13.
    G. Rozenblum, “Regularisation of Secondary Characteristic Classes and Unusual Index Formulas for Operator-Valued Symbols,” in: Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations, Oper. Theory Adv. Appl. 145 (Birkhäuser Verlag, Basel, 2003). pp. 419–437.MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. A. Shubin, “Pseudodifferential Operators in Rn,” Dokl. Akad. Nauk SSSR 196, 316–319 (1971).[Sov. Math. Dokl. 12, 147–151 (1971).MathSciNetGoogle Scholar
  15. 15.
    L. B. Schweitzer, “Spectral Invariance of Dense Subalgebras of Operator Algebras,” Internat. J. Math. 4 (2), 289–317 (1993).zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Yu. Savin and B. Yu. Sternin, “Uniformization of Nonlocal Elliptic Operators and KK-Theory,” Russ. J. Math. Phys. 20 (3), 345–359 (2013).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Peoples’ Friendship University of RussiaMoscowRussia
  2. 2.Institut für Analysis GottfriedWilhelm Leibniz UniversitätHannoverDeutschland

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