On the Index Formula for an Isometric Diffeomorphism

We give an elementary solution of the index problem for elliptic operators associated with the shift operator along the trajectories of an isometric diffeomorphism of a closed smooth manifold. This solution is based on a reduction (which preserves the index) of the operator to an elliptic pseudodifferential operator on a manifold of a higher dimension and an application of an Atiyah--Singer type formula. The final index formula is given in terms of the symbol of the operator on the original manifold.