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 $k_1,k_2,\cdots \in \{1,2,3\}$ so that the sequence of iterates defined by $z_n=X_{k_n}{z}_{n-1}$ does not converge in norm.