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, F, μ, τ), a Fréchet sequence space X with a basis (eₙ)n ≥ 0, and a strongly measurable map T : Ω→ B(X) taking values in a finite set of weighted shifts on X, we study the dynamics of the sequence (T(τⁿ⁻¹ω) T(τω) T(ω))n ≥ 1 for almost every ω∈ Ω. After proving criteria to determine whether this sequence is universal, weakly mixing or mixing for almost every ω∈ Ω, we study some examples on the spaces X = ₚ, X = c₀ and X = H(C) involving two shifts, first in the commuting case and then in the non-commuting one.