Ritt operators and convergence in the method of alternating projections

TL;DR

This paper investigates the convergence behavior of the method of alternating projections in Hilbert spaces, providing new estimates for exponential convergence and demonstrating dense subsets where convergence is arbitrarily fast.

Contribution

It introduces new geometric estimates for convergence rates and shows the existence of dense subsets with arbitrarily fast convergence in the context of Ritt operators.

Findings

New geometric estimates for exponential convergence rates.

Existence of dense subsets with arbitrarily fast polynomial decay.

Strengthening of convergence results via unconditional series.

Abstract

Given $N\ge2$ closed subspaces $M_1,\dotsc, M_N$ of a Hilbert space $X$, let $P_k$ denote the orthogonal projection onto $M_k$, $1\le k\le N$. It is known that the sequence $(x_n)$, defined recursively by $x_0=x$ and $x_{n+1}=P_N\cdots P_1x_n$ for $n\ge0$, converges in norm to $P_Mx$ as $n\to\infty$ for all $x\in X$, where $P_M$ denotes the orthogonal projection onto $M=M_1\cap\dotsc\cap M_N$. Moreover, the rate of convergence is either exponentially fast for all $x\in X$ or as slow as one likes for appropriately chosen initial vectors $x\in X$. 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 $\alpha>0$, a dense subset $X_\alpha$ of $X$ such that $\|x_n-P_Mx\|=o(n^{-\alpha})$ as $n\to\infty$ for all $x\in X_\alpha$. Furthermore, there exists another dense subset $X_\infty$ of $X$ such that, if $x\in X_\infty$, then $\|x_n-P_Mx\|=o(n^{-\alpha})$ as $n\to\infty$ for all $\alpha>0$. 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 $P_M x$ is in fact the limit of a series which converges unconditionally.