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On the index of elliptic operators for the group of dilations. (English. Russian original) Zbl 1240.58017
Sb. Math. 202, No. 10, 1505-1536 (2011); translation from Mat. Sb. 202, No. 10, 99-130 (2011).
The paper deals with index problems for a class of elliptic operators which are finite sums of pseudodifferential operators composed with powers $$T^\alpha$$ of shift operators $$Tu(x)=u(g(x))$$, given by some fixed diffeomorphism $$g$$. More specifically, the authors consider non-isometric diffeomorphisms $$g:x\mapsto qx$$ of the unit sphere $$\mathbb S^m$$ which have two fixed points, the north pole $$N$$ and the south pole $$S$$. Here $$0<q<1$$, and $$x$$ denotes the stereographic coordinates in the complement of $$N$$. The orbit space $$M=\mathbb S_0^m/\mathbb Z$$ of the main stratum $$\mathbb S_0^m=\mathbb S^m\setminus\{N,S\}$$ is compact and diffeomorphic to $$\mathbb S^{m-1}\times \mathbb S^1$$. Using an isomorphism $$I:L^2(\mathbb S_0^m)\to L^2(M,\mathcal E)$$, where $$\mathcal E$$ is a suitable Hilbert bundle, a class of nonlocal pseudodifferential operators is introduced. The operators mentioned above belong to this class. Symbols of nonlocal operators are sections of the endomorphism bundle of $$\mathcal E$$ pulled back to the cosphere bundle $$S^*M$$. A composition formula with compact remainders is shown, as is the Fredholm property of elliptic operators.
In Theorem 2 of the paper it is shown that nonlocal elliptic operators are stably homotopic to (standard) elliptic pseudodifferential operators on $$\mathbb S^m$$, and that their analytical indices are equal. Furthermore, a topological index is defined for nonlocal elliptic operators, and an index formula is proved.
In the last part of the paper, the dependence of the index on the exponent $$s$$ of the Sobolev space $$H^s(\mathbb S^m)$$ is studied.

MSC:
 58J20 Index theory and related fixed-point theorems on manifolds 47A20 Dilations, extensions, compressions of linear operators 35S05 Pseudodifferential operators as generalizations of partial differential operators
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