Strange products of projections

Abstract

Let $H$ be an infinite dimensional Hilbert space. We show that there exist three orthogonal projections $X_1, X_2, X_3$ onto closed subspaces of $H$ such that for every $0\ne z_0\in H$ there exist $k_1, k_2,\dots \in \{1,2,3\}$ so that the sequence of iterates defined by $z_n= X_{k_n} z_{n-1}$ does not converge in norm.