Annals of Global Analysis and Geometry

, Volume 4, Issue 3, pp 349–400 | Cite as

Desuspension of splitting elliptic symbols II

  • Bernhelm Booss
  • Krzystof Wojciechowski
  • Bogdan Bojarski To 
Article

In the second part of our paper we continue the study of elliptic operators which take the form A = GA (a/at + Bt) near a submanifold of codimension 1. The index of the general linear conjugation problem (“cutting and pasting” of elliptic operators) is determined. A thorough analysis of the geometry of Fredholm pairs of subspaces in Hilbert space and especially of the spaces of Cauchy data is undertaken. These methods lead to alternative views of the Calderón projector, the Dirichlet problem, and other local elliptic boundary value problems; views where main results (old and new ones) can be obtained through explicit transparent calculations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Alinhac, S., Non unicité du probléme de Cauchy pour des opérateurs de type principal. II. Ann. of Math. 117 (1983), 77–108.Google Scholar
  2. [2]
    Agranoviĉ, M. S., Elliptic singular intergro-differential operators. Uspehi Mat. Nauk. 20 (1965), no. 5/6, 1–121 (Russian. English translation Russian Math. Surveys).Google Scholar
  3. [3]
    Atiyah, M. F., Bott periodicity and the index of elliptic operators. Quart. J. Math. Oxford (2), 19 (1968), 113–140.Google Scholar
  4. [4]
    Atiyah, M. F., Eigenvalues of the Dirac operator. In: Hirzebruch, F. et al. (eds.), Arbeitstagung Bonn 1984, Proceedings, Lecture Notes on Mathematics 1111, Springer-Verlag, Berlin (West) 1985, 251–260.Google Scholar
  5. [5]
    Atiyah, M. F. and Bott, R. The index problemfor manifolds with boundary. In: Coll. Diff. Analysis, Tata Institute, Bombay, Oxford University Press, Oxford 1964, pp. 175–186.Google Scholar
  6. [6]
    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
  7. [7]
    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
  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. [13]
    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
  12. [14]
    Booss, B. and Rempel, S., Découage etrecollage des opérateurs elliptiques. C. R. Acad. Sci. Paris Sér. I 292 (1981), 711–714.Google Scholar
  13. [15]
    Booss, B. and Rempel, S., Cutting and pasting elliptic operators. Math. Nachr. 109 (1982), 157–194.Google Scholar
  14. [16]
    Booss, B. and Wojciechowski, K., The index of elliptic operators on a mapping torus. C. R. Math. Rep. Acad. Sci. Canada 7 (1985), 97–102.Google Scholar
  15. [17]
    Booss, B. and Wojciechowski, K., Desuspension of splitting elliptic symbols I. Ann. Glob. Analysis and Geometry 3 (1985), 337–383.Google Scholar
  16. [18]
    Calderon, A., Lecture Notes on Pseudo-Differential Operators and Elliptic Boundary Value Problems. I. Consejo Nacional de Investigaciones Cientificas y Tecnicas, Inst. Argentino de Matematica, Buenos Aires 1976.Google Scholar
  17. [19]
    Cordes, H. O., Elliptic Pseudo-Differential Operators — An Abstract Theory. Lecture Notes on Mathematics 756, Springer-Verlag, Berlin (West) 1979.Google Scholar
  18. [20]
    Eskin, G. I., Boundary Value Problems for Elliptic Pseudo-Differential Equations. Moscow 1973.; Amer. Math. Soc. Transl. of Math. Monographs 52, Providence, R. I. 1981.Google Scholar
  19. [21]
    Hörmander, L., Analysis of Linear Partial Differential Operators. III. Springer-Verlag, Berlin (West) 1985.Google Scholar
  20. [22]
    Karoubi, M., K-Theory. Springer-Verlag, Berlin (West) 1978.Google Scholar
  21. [23]
    Kato, T., Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (West) 1976II.Google Scholar
  22. [24]
    Muschelischwili, N. I., Singuläre Integralgleichungen. Akademie-Verlag, Berlin 1965.Google Scholar
  23. [25]
    Palais, R., (ed.), Seminar on the Atiyah-Singer Index Theorem. Ann. of Math. Studies 57: Princeton Univ. Press, Princeton 1965.Google Scholar
  24. [26]
    Pliś, A., A smooth linear elliptic differential equation without any solution in a sphere. Comm. Pure Appl. Math. 14 (1961), 599–617.Google Scholar
  25. [27]
    Rempel, S. and Schulze, B.-W., Index Theory of Elliptic Boundary Problems. Akademie-Verlag, Berlin 1982.Google Scholar
  26. [28]
    Seeley, R., Singular integrals and boundary value problems. Amer. J. Math. 88 (1966), 781–809.Google Scholar
  27. [29]
    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
  28. [30]
    Seeley, R., Elliptic singular integral equations. In: Proc. Symp. Pure Math. vol. 10, Amer. Math. Soc., Providence 1967, pp. 308–315.Google Scholar
  29. [31]
    Shubin, M., Pseudodifferential Operators and Spectral Theory. Nauka, Moscow 1978 (Russian).Google Scholar
  30. [32]
    Solomyak, A., On the Calderon projections. Operator Theory and Function Theory, Leningrad University 1 (1983), 47–55 (Russian).Google Scholar
  31. [33]
    Steenrod, N., The Topology of Fibre Bundles. Princeton University Press, Princeton 1965V.Google Scholar
  32. [34]
    Taylor, M. E., Pseudodifferential Operators. Princeton University Press, Princeton 1981.Google Scholar
  33. [35]
    Vafa, C. and Witten, E., Eigenvalue inequalities for fermions in gauge theories. Comm. Math. Phys. 95, 3 (1984), 257–230.Google Scholar
  34. [36]
    Wojciechowski, K., Spectral Flow and Some Applications to the Index Theory. Doctoral Thesis, Warsaw 1981 (Polish).Google Scholar
  35. [37]
    Wojciechowski, K., Elliptic operators and relative K-homology groups on manifolds with boundary, C. R. Math. Rep. Acad. Sci. Canada 7 (1985), 149–154.Google Scholar
  36. [38]
    Wojciechowski, K., A note on the space of pseudodifferential projections with the same principal symbol. J. Operator Theory 15 (1986), 207–216.Google Scholar
  37. [39]
    Wojciechowski, K., Spectral flow and the general linear conjugation problem, Simon Stevin, 1 (1985), 59–91.Google Scholar
  38. [40]
    Wojciechowski, K., Spectral Asymmetry, Elliptic Boundary Value Problems, and Cutting and Pasting of Elliptic Operators Google Scholar

Copyright information

© Deutscher Verlag der Wissenschaften 1986

Authors and Affiliations

  • Bernhelm Booss
    • 1
  • Krzystof Wojciechowski
    • 2
  • Bogdan Bojarski To 
    • 3
  1. 1.IMFUFARoskilde UniversitetscenterRoskilde
  2. 2.Instytut1Matematyczny Uniwersytet WarszawskiPkiN Warszawa
  3. 3.Instytut Matematyczny Uniwersytet WarszawskiIMFUFA Roskilde UniversitetscenterPkiN Warszawa

Personalised recommendations