# Rates of convergence for alternating projections

### David Seifert (University of Oxford)

Tuesday 15th March, 2016 16:00-17:00 Maths 522

#### Abstract

Given $N\ge2$ closed subspaces $M_1,\dotsc, M_N$ of a Hilbert space $X$, let $P_k$ denote the orthogonal projection onto $M_k$, $1\le k\le N$, and let $P_M$ denote the orthogonal projection onto $M=M_1\cap\dotsc\cap M_N$. Consider the sequence $(x_n)_{n\ge0}$ defined, for $x\in X$, by $x_0=x$ and $x_{n+1}=P_N\cdots P_1x_n$, $n\ge0$. Then, for every $x\in X$,
$$\lim_{n\to\infty}\|x_n-P_Mx\|=0$$
and, depending on the geometric relationship between $M_1,\dotsc, M_N$, the convergence is either slower than any given rate for particular $x\in X$ or exponentially fast for all $x\in X$.

The purpose of this talk is to present an extension of these results based on the theory of (unconditional) Ritt operators. This shows in particular that, even when the convergence can be arbitrarily slow, there is always a dense subset of initial vectors $x\in X$ for which the convergence is faster than polynomial. The talk is based on joint work with C.\ Badea (Lille).