On the index of elliptic operators associated with a diffeomorphism of a manifold

The index of elliptic operators associated with a diffeomorphism of a manifold is calculated. The equality between the indices of the operator under consideration and of a certain boundary value problem on the cylinder is established. A diffeomorphism of a smooth closed manifold induces the shift operator and the symbol of a pseudodifferential operator of order zero is treated as a smooth function. The index formula gives an expression for the index of the boundary value problem in terms of the symbols of the main operator and the operator of boundary conditions. It is shown that a special two term operator is elliptic if the mapping is an isomorphism of bundles. The operator is found to be elliptic and self-adjoint and therefore its nonnegative spectral projection is well defined.