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We present an extension of our earlier work [Ritt operators and convergencein 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 areflexive Banach space. This result is obtained under a condition involving thegrowth of the resolvent, and we also discuss conditions involving the locationand the geometry of the numerical range of the operator. We then apply thegeneral results to some classes of iterative projection methods inapproximation theory, such as the Douglas-Rachford splitting method and, undersuitable geometric conditions either on the ambient Banach space or on theprojection operators, the method of alternating projections.

Quantified asymptotic behaviour of Banach space operators and applications to iterative projection methods