发表机构
Farhangian University; Ferdowsi University of Mashhad（法尔康吉安大学; 马什哈德费尔多西大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究希尔伯特空间中可数无穷多个投影在无限周期选择下的随机乘积，引入无限周期函数类，证明相应乘积序列弱收敛到交集上的投影，并给出技术性例子。
AI 中文摘要
设 $\{P_j\}$ 为希尔伯特空间 $\mathscr{H}$ 的闭子空间 $\mathcal{M}_j$ 上的一列投影。考虑一个无限序列 $P_{i_1}, P_{i_2}, \ldots$，其中每个 $P_{i_n} \in \{P_1, P_2, \ldots\}$，可能按某种顺序重复或随机出现。问题是：在什么条件下，序列 $\{P_{i_n} \cdots P_{i_2} P_{i_1} x\}_{n=1}^{\infty}$ 对每个 $x \in \mathscr{H}$ 强收敛或弱收敛到 $Px$，其中 $P$ 是到交集 $\mathcal{M} = \bigcap_{i=1}^{\infty} \mathcal{M}_i$ 的投影？在本文中，我们给出了关于可数无穷多个投影 $\{P_j\}_{j=1}^{\infty}$ 的随机乘积的一些结果，这些结果引入了无限周期函数的概念。更确切地说，我们引入了一类新的函数 $\sigma \colon \mathbb{N} \to \mathbb{N}$，称为无限周期函数，并严格证明了由 $T_1:= P_{\sigma(1)}$ 和 $T_n:= P_{\sigma(n)} T_{n-1}$（对所有 $n \geq 2$）定义的序列 $\{T_n x\}$（对 $x \in \mathscr{H}$）弱收敛到 $Px$，其中 $P$ 是到 $\bigcap_{j=1}^{\infty} \mathcal{R}(P_j)$ 的投影。我们还提供了一些技术性例子来说明我们的结果。
英文摘要
Let $\{P_j\}$ be a sequence of projections onto the closed subspaces $\mathcal{M}_j$ of a Hilbert space $\mathscr{H}$. Consider an infinite sequence $P_{i_1}, P_{i_2}, \ldots$ with each $P_{i_n} \in \{P_1, P_2, \ldots\}$, possibly repeating in some order or randomly. The question is: Under what conditions does the sequence $\{P_{i_n} \cdots P_{i_2} P_{i_1} x\}_{n=1}^{\infty}$ converge strongly or weakly to $Px$ for every $x \in \mathscr{H}$, where $P$ is the projection onto the intersection $\mathcal{M} = \bigcap_{i=1}^{\infty} \mathcal{M}_i$? In this paper, we present some results concerning random products of countably infinitely many projections $\{P_j\}_{j=1}^{\infty}$ that incorporates the notion of an infinite-periodic function. More precisely, we introduce a new class of functions $σ\colon \mathbb{N} \to \mathbb{N}$, called infinite-periodic functions, and rigorously show that the sequence $\{T_n x\}$ defined by \[ T_1 := P_{σ(1)}\quad \mbox{and} \quad T_n := P_{σ(n)} T_{n-1} \quad \text{for all } n \geq 2, \] for $x \in \mathscr{H}$, converges weakly to $Px$, where $P$ is the projection onto $\bigcap_{j=1}^{\infty} \mathcal{R}(P_j)$. We also provide some technical examples to illustrate our results.
Comments14 pages, to appear in Proc. Amer. Math. Soc
