zbMATH — the first resource for mathematics

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.

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
Full Text: DOI