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.
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References
Alinhac, S., Non unicité du probléme de Cauchy pour des opérateurs de type principal. II. Ann. of Math. 117 (1983), 77–108.
Agranoviĉ, M. S., Elliptic singular intergro-differential operators. Uspehi Mat. Nauk. 20 (1965), no. 5/6, 1–121 (Russian. English translation Russian Math. Surveys).
Atiyah, M. F., Bott periodicity and the index of elliptic operators. Quart. J. Math. Oxford (2), 19 (1968), 113–140.
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.
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.
Atiyah, M. F., Bott, R. and Patodi, V.K., On the heat equation and the index theorem. Invent. Math. 19 (1973), 279–330.
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.
Birman, M. and Solomyak, A., On subspaces which admit pseudodifferential projections. Vestnik Leningrad University 82 no. 1 (1982), 18–25 (Russian).
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).
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.
Booss, B. and Bleecker, D. D., Topology and analysis — The Atiyah-Singer Index Formula and Gauge Theoretic Physics. Springer-Verlag, New York 1985.
Booss, B. and Rempel, S., Découage etrecollage des opérateurs elliptiques. C. R. Acad. Sci. Paris Sér. I 292 (1981), 711–714.
Booss, B. and Rempel, S., Cutting and pasting elliptic operators. Math. Nachr. 109 (1982), 157–194.
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.
Booss, B. and Wojciechowski, K., Desuspension of splitting elliptic symbols I. Ann. Glob. Analysis and Geometry 3 (1985), 337–383.
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.
Cordes, H. O., Elliptic Pseudo-Differential Operators — An Abstract Theory. Lecture Notes on Mathematics 756, Springer-Verlag, Berlin (West) 1979.
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.
Hörmander, L., Analysis of Linear Partial Differential Operators. III. Springer-Verlag, Berlin (West) 1985.
Karoubi, M., K-Theory. Springer-Verlag, Berlin (West) 1978.
Kato, T., Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (West) 1976II.
Muschelischwili, N. I., Singuläre Integralgleichungen. Akademie-Verlag, Berlin 1965.
Palais, R., (ed.), Seminar on the Atiyah-Singer Index Theorem. Ann. of Math. Studies 57: Princeton Univ. Press, Princeton 1965.
Pliś, A., A smooth linear elliptic differential equation without any solution in a sphere. Comm. Pure Appl. Math. 14 (1961), 599–617.
Rempel, S. and Schulze, B.-W., Index Theory of Elliptic Boundary Problems. Akademie-Verlag, Berlin 1982.
Seeley, R., Singular integrals and boundary value problems. Amer. J. Math. 88 (1966), 781–809.
Seeley, R., Complex powers of an elliptic operator. In: Proc. Symp. Pure Math. vol. 10, Amer. Math. Soc., Providence 1967, pp. 288–307.
Seeley, R., Elliptic singular integral equations. In: Proc. Symp. Pure Math. vol. 10, Amer. Math. Soc., Providence 1967, pp. 308–315.
Shubin, M., Pseudodifferential Operators and Spectral Theory. Nauka, Moscow 1978 (Russian).
Solomyak, A., On the Calderon projections. Operator Theory and Function Theory, Leningrad University 1 (1983), 47–55 (Russian).
Steenrod, N., The Topology of Fibre Bundles. Princeton University Press, Princeton 1965V.
Taylor, M. E., Pseudodifferential Operators. Princeton University Press, Princeton 1981.
Vafa, C. and Witten, E., Eigenvalue inequalities for fermions in gauge theories. Comm. Math. Phys. 95, 3 (1984), 257–230.
Wojciechowski, K., Spectral Flow and Some Applications to the Index Theory. Doctoral Thesis, Warsaw 1981 (Polish).
Wojciechowski, K., Elliptic operators and relative K-homology groups on manifolds with boundary, C. R. Math. Rep. Acad. Sci. Canada 7 (1985), 149–154.
Wojciechowski, K., A note on the space of pseudodifferential projections with the same principal symbol. J. Operator Theory 15 (1986), 207–216.
Wojciechowski, K., Spectral flow and the general linear conjugation problem, Simon Stevin, 1 (1985), 59–91.
Wojciechowski, K., Spectral Asymmetry, Elliptic Boundary Value Problems, and Cutting and Pasting of Elliptic Operators
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Booss, B., Wojciechowski, K. & To Bogdan, B. Desuspension of splitting elliptic symbols II. Ann Glob Anal Geom 4, 349–400 (1986). https://doi.org/10.1007/BF00128052
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DOI: https://doi.org/10.1007/BF00128052