Convergence of Random Products of Countably Infinitely Many Projections

Let r ∈ N∪\∞\ be a fixed number and let Pⱼ\,\, (1 ≤ j≤ r ) be the projection onto the closed subspace Mⱼ of H. We are interested in studying the sequence Pi₁, Pi₂, … ∈₁, …, Pᵣ\. A significant problem is to demonstrate conditions under which the sequence Πₙ⋯ Pi₂Pi₁x\ₙ₌₁^∞ converges strongly or weakly to $Px$ for any x, where P is the projection onto the intersection M=M₁∩ … ∩ Mᵣ. Several mathematicians have presented their insights on this matter since von Neumann established his result in the case of $r=2$. In this paper, we give an affirmative answer to a question posed by M. Sakai. We present a result concerning random products of countably infinitely many projections (the case r=∞) incorporating the notion of pseudo-periodic function.