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Index of elliptic operators for diffeomorphisms of manifolds. (English) Zbl 1320.58010
The authors deal with operators that are obtained by iterates of a given diffeomorphism of a closed manifold \(M\) composed with pseudo-differential operators on \(M\) and finite sums of these. The algebra of symbols of those operators is noncommutative. An elliptic theory for such operators is developped. In particular, a notion of being elliptic is defined which implies Fredholmness and regularity properties of kernel and cokernel. An index formula in terms of topological invariants and the symbol of the operator is presented.

MSC:
58J05 Elliptic equations on manifolds, general theory
19K56 Index theory
46L85 Noncommutative topology
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
58J40 Pseudodifferential and Fourier integral operators on manifolds
58J20 Index theory and related fixed-point theorems on manifolds
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[1] A. B. Antonevich, Elliptic pseudodifferential operators with a finite group of shifts. Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 663-675; English transl. Math. USSR-Izv. 7 (1973), 661-674. · Zbl 0283.47034 · doi:10.1070/IM1973v007n03ABEH001968
[2] A. Antonevich, M. Belousov, and A. Lebedev, Functional differential equations . II. C - applications. Part 1: Equations with continuous coefficients. Pitman Monogr. Surveys Pure Appl. Math. 94, Longman, Harlow 1998. · Zbl 0936.35207
[3] A. Antonevich and A. Lebedev, Functional-differential equations . I. C -theory. Pitman Monogr. Surveys Pure Appl. Math. 70, Longman Scientific & Technical, Harlow 1994. · Zbl 0799.34001
[4] V. I. Arnold, Mathematical methods of classical mechanics . 2nd ed., Graduate Texts in Math. 60, 2nd ed., Springer-Verlag, New York 1989. · Zbl 0386.70001
[5] M. F. Atiyah, K -theory. 2nd ed., Addison-Wesley Publishing Company, Redwood City, CA, 1989. · Zbl 0676.55006
[6] M. F. Atiyah and F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds. Bull. Amer. Math. Soc. 65 (1959), 276-281. · Zbl 0142.40901 · doi:10.1090/S0002-9904-1959-10344-X
[7] M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian ge- ometry. III. Math. Proc. Cambridge Philos. Soc. 79 (1976), 71-99. · Zbl 0325.58015 · doi:10.1017/S0305004100052105
[8] M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds. Bull. Amer. Math. Soc. 69 (1963), 422-433. · Zbl 0118.31203 · doi:10.1090/S0002-9904-1963-10957-X
[9] P. Baum and R. G. Douglas, K homology and index theory. In Operator algebras and applications , Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math. 38, Amer. Math. Soc., Providence, R.I., 1982, 117-173. · Zbl 0532.55004
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