Do Projections Stay Close Together?

We estimate the rate of convergence of products of projections on lines in R. Consider the orbit of a point under any sequence of orthogonal projections on K lines in R. Assume that the sum of the squares of the distances of the consecutive iterates is less than ε. We show that if ε tends to zero, then the diameter of the orbit tends to zero uniformly for all families L of a fixed number K of lines. We relate this result to questions concerning convergence of products of projections on finite families of closed subspaces of l2. Introduction Let K be a fixed natural number and let L be a family of K affine subspaces of R. Let z ∈ R and k1, k2, · · · ∈ {1, 2, . . . , K} be arbitrary. Consider the sequence of projections z1 = Pk1z, zn = Pknzn−1, where Pk denotes the orthogonal projection on the k-th space in L. The orbit {zi} is always bounded according to [ADW], [BGP], and [Me]. In this note we consider an additional constraint on the distances of the consecutive iterates, namely that