Linear dynamics of random products of operators

Abstract

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 $\mathbb{T}$.