Linear dynamics of random products of weighted shifts

TL;DR

This paper investigates the long-term behavior of random products of weighted shift operators on various sequence spaces, establishing criteria for universality, mixing, and weak mixing in a probabilistic dynamical setting.

Contribution

It introduces new criteria to determine the dynamics of random weighted shift products and analyzes their behavior on classical sequence spaces, including commuting and non-commuting cases.

Findings

Criteria for universality, weak mixing, and mixing established.

Examples on ll_p, c_0, and H(b) spaces analyzed.

Differentiates dynamics in commuting and non-commuting cases.

Abstract

The aim of this article is to study the dynamics of random products of weighted shifts on a separable Fr\'echet sequence space. That is, given a measure-preserving dynamical system $(\Omega, \mathcal{F}, \mu, \tau)$, a Fr\'echet sequence space $X$ with a basis $(e_n)_{n \geq 0}$, and a strongly measurable map $T : \Omega \to \mathcal{B}(X)$ taking values in a finite set of weighted shifts on $X$, we study the dynamics of the sequence $(T(\tau^{n-1}\omega) \dotsm T(\tau \omega) T(\omega))_{n \geq 1}$ for almost every $\omega \in \Omega$. After proving criteria to determine whether this sequence is universal, weakly mixing or mixing for almost every $\omega \in \Omega$, 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.