Ritt operators and convergence in the method of alternating projections

Given N≥2 closed subspaces M1,…,MN of a Hilbert space X, let Pk denote the orthogonal projection onto Mk, 1≤k≤N. It is known that the sequence (xn), defined recursively by x0=x and xn+1=PN⋯P1xn for n≥0, converges in norm to PMx as n→∞ for all x∈X, where PM denotes the orthogonal projection onto M=M1∩…∩MN. Moreover, the rate of convergence is either exponentially fast for all x∈X or as slow as one likes for appropriately chosen initial vectors x∈X. We give a new estimate in terms of natural geometric quantities on the rate of convergence in the case when it is known to be exponentially fast. More importantly, we then show that even when the rate of convergence is arbitrarily slow there exists, for each real number α>0, a dense subset Xα of X such that ‖xn−PMx‖=o(n−α) as n→∞ for all x∈Xα. Furthermore, there exists another dense subset X∞ of X such that, if x∈X∞, then ‖xn−PMx‖=o(n−α) as n→∞ for allα>0. These latter results are obtained as consequences of general properties of Ritt operators. As a by-product, we also strengthen the unquantified convergence result by showing that PMx is in fact the limit of a series which converges unconditionally.