Ritt operators and convergence in the method of alternating projections

Abstract

Given $N\ge 2$ closed subspaces $M_1,\dots ,M_N$ 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 $x_{n+1}=P_N\cdots P_1{x}_n$ for $n\ge 0$, converges in norm to $P_{M}x$ as $n\to \infty$ for all $x\in X$, where $P_M$ denotes the orthogonal projection onto $M=M_1\cap \dots \cap M_N$. 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 $\Vert x_n-P_{M}x\Vert =o(n^{-\alpha })$ 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 $\Vert x_n-P_{M}x\Vert =o(n^{-\alpha })$ 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.

To appear in Journal of Approximation Theory

References in corpus (1)

• Rates of decay in the classical Katznelson-Tzafriri theorem

Cited by in corpus (6)

• When products of projections diverge
• Some developments around the Katznelson-Tzafriri theorem
• Optimal rates of decay in the Katznelson-Tzafriri theorem for operators on Hilbert spaces
• Non-optimality of the Greedy Algorithm for subspace orderings in the method of alternating projections
• On the optimal error bound for the first step in the method of cyclic alternating projections
• The Optimal Error Bound for the Method of Simultaneous Projections