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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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##### References:
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