  We present an extension of our earlier work [Ritt operators and convergence
in the method of alternating projections, J. Approx. Theory, 205:133-148, 2016]
by proving a general asymptotic result for orbits of an operator acting on a
reflexive Banach space. This result is obtained under a condition involving the
growth of the resolvent, and we also discuss conditions involving the location
and the geometry of the numerical range of the operator. We then apply the
general results to some classes of iterative projection methods in
approximation theory, such as the Douglas-Rachford splitting method and, under
suitable geometric conditions either on the ambient Banach space or on the
projection operators, the method of alternating projections.
