Quantified asymptotic behaviour of Banach space operators and applications to iterative projection methods
For researchers in operator theory and approximation, this provides a unified asymptotic framework, but the results are incremental extensions of prior work.
The paper extends earlier work on Ritt operators to prove a general asymptotic result for orbits of operators on reflexive Banach spaces, with applications to iterative projection methods like Douglas-Rachford splitting and alternating projections.
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.