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

TL;DR

This paper extends previous work on Banach space operators by establishing a general asymptotic behavior result for operator orbits, with applications to iterative projection methods like Douglas-Rachford and alternating projections.

Contribution

It introduces a new asymptotic analysis framework based on resolvent growth and numerical range geometry, applicable to various iterative projection algorithms.

Findings

Established a general asymptotic behavior theorem for operator orbits.

Applied results to the Douglas-Rachford splitting method.

Analyzed conditions for convergence of alternating projections.

Abstract

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.