Linear dynamics of random products of weighted shifts

The aim of this article is to study the dynamics of random products of weighted shifts on a separable Fréchet sequence space. That is, given a measure-preserving dynamical system $(\mathrm{\Omega },\mathcal{F},\mu ,\tau )$, a Fréchet sequence space $X$ with a basis $(e_n{)}_{n\ge 0}$, and a strongly measurable map $T:\mathrm{\Omega }\to \mathcal{B}(X)$ taking values in a finite set of weighted shifts on $X$, we study the dynamics of the sequence $(T({\tau }^{n-1}\omega )\cdots T(\tau \omega )T(\omega ){)}_{n\ge 1}$ for almost every $\omega \in \mathrm{\Omega }$. After proving criteria to determine whether this sequence is universal, weakly mixing or mixing for almost every $\omega \in \mathrm{\Omega }$, we study some examples on the spaces $X={\mathrm{\ell }}_p$, $X=c_0$ and $X=H(\mathbb{C})$ involving two shifts, first in the commuting case and then in the non-commuting one.