Ritt operators and convergence in the method of alternating projections

Given N≥2N≥2 closed subspaces M1,…,MNM1,…,MN of a Hilbert space XX, let PkPk denote the orthogonal projection onto MkMk, 1≤k≤N1≤k≤N. It is known that the sequence (xn)(xn), defined recursively by x0=xx0=x and xn+1=PN⋯P1xnxn+1=PN⋯P1xn for n≥0n≥0, converges in norm to PMxPMx as n→∞n→∞ for all x∈Xx∈X, where PMPM denotes the orthogonal projection onto M=M1∩…∩MNM=M1∩…∩MN. Moreover, the rate of convergence is either exponentially fast for all x∈Xx∈X or as slow as one likes for appropriately chosen initial vectors x∈Xx∈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>0α>0, a dense subset XαXα of XX such that ‖xn−PMx‖=o(n−α)‖xn−PMx‖=o(n−α) as n→∞n→∞ for all x∈Xαx∈Xα. Furthermore, there exists another dense subset X∞X∞ of XX such that, if x∈X∞x∈X∞, then ‖xn−PMx‖=o(n−α)‖xn−PMx‖=o(n−α) as n→∞n→∞ for all  α>0α>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 PMxPMx is in fact the limit of a series which converges unconditionally.