Convergence of Random Products of Countably Infinitely Many Projections

2024/05/08 by Rasoul Eskandari, Mohammad Sal Moslehian, Eskandari, Rasoul +1
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #Mathematical Dynamics and Fractals #Advanced Topology and Set Theory

paper · pdf · doi:10.48550/arxiv.2405.04848

Abstract

Let r ∈ ℕ∪\∞\ be a fixed number and let P_j (1 ≤ j≤ r ) be the projection onto the closed subspace M_j of \mathscrH. We are interested in studying the sequence P_i1, P_i2, … ∈\P₁, …, P_r\. A significant problem is to demonstrate conditions under which the sequence \P_in⋯ P_i2P_i1x\_n=1^∞ converges strongly or weakly to Px for any x∈\mathscrH, where P is the projection onto the intersection M=M₁∩ … ∩ M_r. 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.

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