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Nonlocal elliptic operators for compact Lie groups. (English. Russian original) Zbl 1213.47012
Dokl. Math. 81, No. 2, 258-261 (2010); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 431, No. 4, 457-460 (2010).
The paper considers the class of nonlocal operators corresponding to shifts along orbits under the action of a compact Lie group on a smooth manifold. Suppose that a compact Lie group \(G\) acts on a smooth closed manifold \(M\). Let \(T^*_{G}M:=\{(x, \xi) \in T^*M \mid \langle\xi, X_{M}(x) = 0\;\forall X \in \mathcal(G)\}\) and \(S^*_{G}M\) be the subspace consisting of covectors with unit length in \(T^*_{G}M\). Then \(G\) acts on \(T^*_{G}M\) and \(S^*_{G}M\) by differentials. Denote the crossed product of the algebra \(C(S^*_{G}M)\) and \(G\) corresponding to the action of \(G\) by \(C(S^*_{G}M) \rtimes G\).
The paper introduces a definition of ellipticity for nonlocal operators. Namely, a nonlocal operator on \(L^{2}(M, {\mathbb{C}}^{N})\) is called elliptic if and only if its principal symbol is invertible in the algebra Mat\(_{N}( C(S^*_{G}M) \rtimes G )^{+}\). The main results of the paper are Theorems 1,2 which assert that elliptic nonlocal operators are Fredholm and give an index formula for these operators in topological terms.

MSC:
47A53 (Semi-) Fredholm operators; index theories
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