On the entangled ergodic theorem

Abstract

k} is a surjective map. We show that, on general Banach spaces and without any restriction on the partition α, the above averages converge strongly as N → ∞ under some quite weak compactness assumptions on the operators T j and A j . A formula for the limit based on the spectral analysis of the operators T j and the continuous version of the result are presented as well.

Almost weak polynomial stability of operators

“…We now show that under certain compactness asssumptions, the averages (2) converge in the strong operator topology. This is a generalization of the results in Eisner, K-K [9] an eigenvector to some unimodular eigenvalue λ ∈ Γ. Then U p α(1) (n α(1) ) A 1 U p α(2) (n α(2) ) A 2 • • • U p α(k) (n α(k) ) A k U − k j=1 p α(j) (n α(j) ) x = 1 N r N n 1 ,...,nr=1…”

“…In this section we apply the previous results to the setting of polynomial multiple ergodic averages, much in the vein of Eisner, Kunszenti-Kovács [9].…”

“…depending on k ∈ N + , showing that in contrast to the commutative case, one cannot expect convergence in general if k ≥ 3 . On the other hand it is shown in Section 4 of [9] that for every von Neumann dynamical system there is a large class (see below) K depending on the system such that the multiple ergodic averages converge strongly whenever a 1 , . .…”

“…This is now a form where each power is a single polynomial, and the induction arguments from the proof of Theorem 3 in [9] can be applied to show convergence.…”

Pointwise entangled ergodic theorems for Dunford–Schwartz operators

“…The operators A i are acting as transitions between the actions of the operators T i , that iteratively govern the dynamics, whereas the entanglement map α provides a coupling between the stages. Further papers on the subject initially focused on strong convergence of these Cesàro averages, see Liebscher [20], Fidaleo [10,11,12] and Eisner, K.-K. [8]. In Eisner, K.-K. [9] and K.-K. [19], attention was turned to pointwise almost everywhere convergence in the context of the T i 's being operators on function spaces E = L p (X, µ) (1 ≤ p < ∞), where (X, µ) is a standard probability space (i.e.…”