An index formula for nonlocal operators corresponding to a diffeomorphism of a manifold

The index problem for nonlocal elliptic operators representable as a finite sum of the form of pseudodifferential operators is studied. Index formulas for operators are obtained for special diffeomorphisms and the index of operators corresponding to a linear shift on the torus are calculated. Topological invariants of nonlocal elliptic operators for diffeomorphism of general form are constructed and the analytic index of an operator in terms of these invariants is expressed. An operator is said to be elliptic if, for this operator, there exists an inverse symbol with finitely many nonzero components. The Chern character on the K-group of crossed products with the group is defined. For an elliptic operator, the Chern character with values in the Haefliger cohomology are defined.