# zbMATH — the first resource for mathematics

Index problem for elliptic operators associated with a diffeomorphism of a manifold and uniformization. (English. Russian original) Zbl 1248.47046
Dokl. Math. 84, No. 3, 846-849 (2011); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 441, No. 5, 593-596 (2011).
Let $$M$$ be a smooth manifold on which a smooth isometric diffeomorphism $$g: M\to M$$ is given. The powers of $$g$$ generate an action of the group $$\mathbb Z$$ on $$M.$$ The authors consider the differential operator with shift on $$M$$ defined by $D=\sum D_kT^k: C^\infty(M)\to C^\infty(M),$ where $$T^ku(x)=u(g^k(x))$$ and $$D_k$$ are differential operators. The operator $$T$$ is Fredholm under an appropriate ellipticity condition, and the main purpose of the note under review is to calculate its index.

##### MSC:
 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) 58J20 Index theory and related fixed-point theorems on manifolds 58B15 Fredholm structures on infinite-dimensional manifolds
Full Text:
##### References:
 [1] A. Antonevich and A. Lebedev, Functional Differential Equations, I: C*-Theory (Longman, Harlow, 1994). · Zbl 0799.34001 [2] V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin, Elliptic Theory and Noncommutative Geometry (Birkhäuser, Basel, 2008). [3] P. Baum and A. Connes, A Fête of Topology (Academic, Boston, Mass., 1988). [4] G. Luke, J. Differ. Equations 12, 566–589 (1972). · Zbl 0238.35077 · doi:10.1016/0022-0396(72)90026-5 [5] M. F. Atiyah and I. M. Singer, Ann. Math. 87, 484–530 (1968). · Zbl 0164.24001 · doi:10.2307/1970715 [6] V. E. Nazaikinskii, A. Yu. Savin, B.-W. Schulze, and B. Yu. Sternin, Elliptic Theory on Singular Manifolds (CRC Press, Boca Raton, Fla., 2005). [7] G. Rozenblum, Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations (Birkhäuser, Basel, 2003), pp. 419–437. · Zbl 1161.58311 [8] L. Hörmander, The Analysis of Linear Partial Differential Operators (Springer-Verlag, Berlin, 1985), vol. 3.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.