Annals of Global Analysis and Geometry

, Volume 3, Issue 3, pp 337–383 | Cite as

Desuspension of splitting elliptic symbols I

  • Booss Bernhelm 
  • Wojciechowski Krzysztof 

This paper provides an algorithm for the conversion of the index of an elliptic first-order differential operator A on the torus Y×S1 into the index of a canonically associated elliptic pseudo-differential operator Q and Y . It is supposed that Y is a closed smooth manifold and that A “splits” into ∂/∂t + Bt , where {Bt} is a family of self-adjoint elliptic operators on Y satisfying the periodicity condition B1 = g B0 g−1 for some unitary automorphism g . Then it will be shown that the operator Q ( the “desuspension” of A ) can be written down explicity in the form Q = P+ −gP where P+ are projections onto the space of Cauchy data. In the second part , applications are given for the calculation of the index of the general linear conjugation problem (“cutting and pasting” of elliptic operators) , and the intimate interrelations between the related procedures of algebraic topology, spectral theory and functional analysis are explained . Generalizations in various directions are indicated.


Differential Operator Group Theory Periodicity Condition Spectral Theory Elliptic Operator 
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  1. [1]Atiyah, M. F. and Bott, R., The index problem for manifolds with boundary. In:Coll. Diff. Analysis, Tata Institute, Bombay, Oxford University Press, Oxford 1964, pp. 175–186.Google Scholar
  2. [2]Atiyah, M. F., Bott, R. and Patodi, V. K., On the heat equation and the index theorem.Invent. Math. 19 (1973), 279–330.Google Scholar
  3. [3]Atiyah, M. F., Patodi, V. K. and Singer, I. M., Spectral asymmetry and Riemannian geometry. I.Math. Proc. Camb. Phil. Soc. 77 (1975), 43–69.Google Scholar
  4. [4]Atiyah, M. F., Patodi, V. K. and Singer, I. M., Spectral asymmetry and Riemannian geometry. III.Math. Proc. Camb. Phil. Soc. 79 (1976), 71–99.Google Scholar
  5. [5]Atiyah, M. F. and Singer, I. M., Index theoryfor skew-adjoint Fredholm operators.Publ. Math. Inst. Hautes Etudes Sci. no. 37 (1969), 5–26.Google Scholar
  6. [6]Atiyah, M. F. and Singer, I. M., The index of elliptic operators. IV.Ann. of Math. 93 (1971), 119–138.Google Scholar
  7. [7]Bessaga, Cz. and Pelczynski, A.,Selected Topics in Infinite-Dimensional Topology. Polish Scientific Publishers, Warsaw 1975.Google Scholar
  8. [8]Birman, M. and Solomyak, A., On subspaces which admit pseudodifferential projections.Vestnik Leningrad University 82 no. 1 (1982), 18–25 (Russian).Google Scholar
  9. [9]Bojarski, B., The abstract linear conjugation problem and Fredholm pairs of subspaces. In:In Memoriam I. N. Vekua, Tbilisi University, Tbilisi 1979, pp. 45–60 (Russian).Google Scholar
  10. [10]Bojarski, B., Connections between complex and global analysis — Some analytical and geometrical aspects of the Riemann-Hilbert transmission problem. In: Lanckau, E. and Tutschke, E. (eds.),Complex Analysis — Methods, Trends, and Applications, Akademie-Verlag, Berlin 1983, pp. 97–110.Google Scholar
  11. [11]Booss, B., Eindeutige Fortsetzbarket für elliptische Operatoren und ihre formal Adjungierten. Bonn 1965 (multiplied).Google Scholar
  12. [12]Booss, B. and Bleecker, D. D.,Topology and Analysis — The Atiyah-Singer Index Formula and Gauge Theoretic Physics. Springer-Verlag, New York 1985.Google Scholar
  13. [13]Booss, B. and Rempel, S., Decoupage et recollage des operateurs elliptiques.C. R. Acad. Sci. Paris Ser. I 292 (1981), 711–714.Google Scholar
  14. [14]Booss, B. and Rempel, S., Cutting and pasting of elliptic operators.Math. Nachr. 109 (1982), 157–194.Google Scholar
  15. [15]Booss, B. and Wojciechowski, K., The index of elliptic operators on a mapping torus.Math. Reports of the Royal Society of Canada (forthcoming).Google Scholar
  16. [16]Calderon, A.,Lecture Notes on Pseudo-Differential Operators and Elliptic Boundary Value Problems. I. Inst. Argentino de Matematica, Buenos Aires 1976.Google Scholar
  17. [17]Courant, R. and Hilbert, D.,Methoden der Mathematischen Physik I. Springer-Verlag, Berlin (West) 1963III.Google Scholar
  18. [18]Friedrichs, K. O.,Spectral Theory of Operators in Hilbert Space. Springer-Verlag, Berlin (West) 1973.Google Scholar
  19. [19]Fujii, K., A representation of complex K-groups by means of a Banach algebra.Mem. Fac. Sci. Kyushu Univ. Ser A 32 (1978), no. 2, 255–265.Google Scholar
  20. [20]Guillemin, V., Lectures on spectral theory of elliptic operators.Duke Math. J. 44 (1977), 485–518.Google Scholar
  21. [21]Hirzebruch, F. and Scharlau, W.,Einführung in die Funktionalanalysis. B-I-Hochschultaschenbücher vol 296, Mannheim 1971.Google Scholar
  22. [22]Hörmander, L., Pseudo-differential operators and non-elliptic boundary problems.Ann. of Math. 83 (1966), 129–209.Google Scholar
  23. [23]Karoubi, M.,K-Theory. Springer-Verlag, Berlin (West) 1978.Google Scholar
  24. [24]Kato, T.,Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (West) 1976II.Google Scholar
  25. [25]Milnor, J., On spaces having the homotopy type of a CW-complex.Trans. Amer. Math. Soc. 90 (1957), 272–280.Google Scholar
  26. [26]Muschelischwili, N. I.,Singulare Integralgleichungen. Akademie-Verlag, Berlin 1965.Google Scholar
  27. [27]Palais, R., (ed.),Seminar on the Atiyah-Singer Index Theorem. Ann. of Math. Studies 57, Princeton Univ. Press, Princeton 1965.Google Scholar
  28. [28]Palais, R., On the homotopy groups of certain groups of operators.Topology 3 (1965), 271–279.Google Scholar
  29. [29]Plis, A., A smooth linear elliptic differential equation without any solution in a sphere.Comm. Pure Appl. Math. 14 (1961), 599–617.Google Scholar
  30. [30]Seeley, R., Singular integrals and boundary value problems.Amer. J. Math. 88 (1966), 781–809.Google Scholar
  31. [31]Seeley, R., Complex powers of an elliptic operator. In:Proc. Symp. Pure Math. vol. 10, Amer. Math. Soc., Providence 1967, pp. 288–307.Google Scholar
  32. [32]Seeley, R., Elliptic singular integral equations. In:Proc. Symp. Pure Math. vol. 10, Amer. Math. Soc., Providence 1967, pp. 308–315.Google Scholar
  33. [33]Shubin, M.,Pseudodifferential Operators and Spectral Theory. Nauka, Moscow 1978 (Russian).Google Scholar
  34. [34]Steenrod, N.,The Topology of Fibre Bundles. Princeton University Press, Princeton 1965V.Google Scholar
  35. [35]Treves, F.,Introduction to Pseudodifferential and Fourier Integral Operators. I. Plenum Press, New York 1980.Google Scholar
  36. [36]Wodzicki, M., Spectral asymetry and zeta-functions.Invent. Math. 66 (1982), 115–135.Google Scholar
  37. [37]Wojciechowski, K.,Spectral Flow and Some Applications to the Index Theory. Doctoral Thesis, Warsaw 1981 (Polish).Google Scholar

Copyright information

© VEB Deutscher Verlag der Wissenschaften 1985

Authors and Affiliations

  • Booss Bernhelm 
    • 1
  • Wojciechowski Krzysztof 
    • 2
  1. 1.Institut for Studiet af Matematik og FysikRoskilde UniversitetscenterRoskilde
  2. 2.Instytut MatematycznyUniwersytet WarszawskiPKiN Warszawa

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