标题： 算子随机乘积的线性动力学 显示英文标题

标题： Linear dynamics of random products of operators

摘要： 我们研究随机序列$(T_n(.))_{n \geq 1}$的线性动力学，该序列由算子$T_n(\omega) = T(\tau^{n-1}\omega) \dotsm T(\tau \omega) T(\omega), n \geq 1$组成。 这些乘积依赖于概率空间$(\mathbb{T}, m)$上的遍历保测变换$\tau : \mathbb{T} \to \mathbb{T}$，以及一个强可测映射$T : \mathbb{T} \to \mathcal{B}(X)$，其中$X$是一个可分的弗雷歇空间。 我们将关注当$T(\omega)$对于每个$\omega \in A_1$等于算子$T_1$在$X$上，对于每个$\omega \in A_2$等于算子$T_2$在$X$上的情况，其中$A_1, A_2$是$[0,1)$的两个不相交的 Borel 子集，使得$A_1 \cup A_2 = [0,1)$和$m(A_k) > 0$对$k = 1,2$成立。 更准确地说，我们将关注算子$T_1$和$T_2$是 Hardy 空间$H^2(\mathbb{D})$上乘法算子的伴随算子的情况，以及算子$T_1$和$T_2$是全函数空间上微分算子的指数型整函数的情况。 最后，我们将研究一个随机乘积 $T_n(\omega)$的线性动力学情况，其中算子 $T(\tau^i \omega), i \geq 0$不对易。我们将特别重视遍历变换为无理旋转或 $\mathbb{T}$上的二倍映射的情况。 显示英文摘要

摘要： We study the linear dynamics of the random sequence $(T_n(.))_{n \geq 1}$ of the operators $T_n(\omega) = T(\tau^{n-1}\omega) \dotsm T(\tau \omega) T(\omega), n \geq 1$. These products depend on an ergodic measure-preserving transformation $\tau : \mathbb{T} \to \mathbb{T}$ on the probability space $(\mathbb{T}, m)$ and on a strongly measurable map $T : \mathbb{T} \to \mathcal{B}(X)$, where $X$ is a separable Fr\'echet space. We will be focusing on the case where $T(\omega)$ is equal to an operator $T_1$ on $X$ for every $\omega \in A_1$ and equal to an operator $T_2$ on $X$ for every $\omega \in A_2$, where $A_1, A_2$ are two disjoint Borel subsets of $[0,1)$ such that $A_1 \cup A_2 = [0,1)$ and $m(A_k) > 0$ for $k = 1,2$. More precisely, we will be focusing on the case where the operators $T_1$ and $T_2$ are adjoints of multiplication operators on the Hardy space $H^2(\mathbb{D})$, as well as the case where $T_1$ and $T_2$ are entire functions of exponential type of the derivation operator on the space of entire functions. Finally, we will study the linear dynamics of a case of a random product $T_n(\omega)$ for which the operators $T(\tau^i \omega), i \geq 0$, do not commute.We will give particular importance to the case where the ergodic transformation is an irrational rotation or the doubling map on $\mathbb{T}$.