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 $(Ω, \\mathcal{F}, μ, τ)$, a Fréchet sequence space $X$ with a basis $(e_n)_{n \\geq 0}$, and a strongly measurable map $T : Ω\\to \\mathcal{B}(X)$ taking values in a finite set of weighted shifts on $X$, we study the dynamics of the sequence $(T(τ^{n-1}ω) \\dotsm T(τω) T(ω))_{n \\geq 1}$ for almost every $ω\\in Ω$. After proving criteria to determine whether this sequence is universal, weakly mixing or mixing for almost every $ω\\in Ω$, we study some examples on the spaces $X = \\ell_p$, $X = c_0$ and $X = H(\\mathbb{C})$ involving two shifts, first in the commuting case and then in the non-commuting one.