Let r ∈ N∪{∞} be a fixed number and let P j (1 ≤ j ≤ r ) be the projection onto the closed subspace M j of H . We are interested in studying the sequence P i 1 , P i2 , . . . ∈ {P 1 , . . . , P r }. A significant problem is to demonstrate conditions under which the sequence {P in · · · P i 2 P i 1 x}∞ n=1 converges strongly or weakly to P x for any x ∈ H , 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. © 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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