Convergence of Random Products of Projections Under Infinite-Periodic Selections
Abstract
Let {Pj} be a sequence of projections onto the closed subspaces ℳj of a Hilbert space ℋ. Consider an infinite sequence Pi1, Pi2, … with each Pin ∈ {P1, P2, …}, possibly repeating in some order or randomly. The question is: Under what conditions does the sequence {Pin ⋯ Pi2 Pi1 x}n=1∞ converge strongly or weakly to Px for every x ∈ ℋ, where P is the projection onto the intersection ℳ = ⋂i=1∞ ℳi? In this paper, we present some results concerning random products of countably infinitely many projections {Pj}j=1∞ that incorporates the notion of an infinite-periodic function. More precisely, we introduce a new class of functions σ: ℕ → ℕ, called infinite-periodic functions, and rigorously show that the sequence {Tn x} defined by T1 := Pσ(1) and Tn := Pσ(n) Tn−1 for all n ≥ 2, for x ∈ ℋ, converges weakly to Px, where P is the projection onto ⋂j=1∞ ℛ(Pj). We also provide some technical examples to illustrate our results.
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BibTeX
@misc{eskandari2026convergence,
title = {{Convergence of Random Products of Projections Under Infinite-Periodic Selections}},
author = {Rasoul Eskandari and Mohammad Sal Moslehian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13957},
eprint = {2609.13957},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13957},
}
RIS
TY - UNPB
TI - Convergence of Random Products of Projections Under Infinite-Periodic Selections
AU - Rasoul Eskandari
AU - Mohammad Sal Moslehian
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13957
UR - https://arxiv.org/abs/2609.13957
AB - Let {P_j} be a sequence of projections onto the closed subspaces ℳ_j of a Hilbert space ℋ. Consider an infinite sequence P_i_1, P_i_2, … with each P_i_n ∈ {P_1, P_2, …}, possibly repeating in some order or randomly. The question is: Under what conditions does the sequence {P_i_n ⋯ P_i_2 P_i_1 x}_n=1^∞ converge strongly or weakly to Px for every x ∈ ℋ, where P is the projection onto the intersection ℳ = ⋂_i=1^∞ ℳ_i? In this paper, we present some results concerning random products of countably infinitely many projections {P_j}_j=1^∞ that incorporates the notion of an infinite-periodic function. More precisely, we introduce a new class of functions σ: ℕ → ℕ, called infinite-periodic functions, and rigorously show that the sequence {T_n x} defined by T_1 := P_σ(1) and T_n := P_σ(n) T_n−1 for all n ≥ 2, for x ∈ ℋ, converges weakly to Px, where P is the projection onto ⋂_j=1^∞ ℛ(P_j). We also provide some technical examples to illustrate our results.
ER -