On the entangled ergodic theorem

Abstract

Let $U$ be a unitary operator acting on the Hilbert space H, and $\alpha :\{1,...,m\}\mapsto \{1,...,k\}$ a partition of the set $\{1,...,m\}$. We show that the ergodic average $$\frac{1}{N^k}\sum _{n_1,...,n_k=0}^{N-1}{U}^{n_{\alpha (1)}}{A}_1{U}^{n_{\alpha (2)}}...U^{n_{\alpha (m-1)}}{A}_{m-1}{U}^{n_{\alpha (m)}}$$ converges in the weak operator topology if the $A_j$ belong to the algebra of all the compact operators on H. We write esplicitely the formula for these ergodic averages in the case of pair--partitions. Some results without any restriction on the operators $A_j$ are also presented in the almost periodic case.

arXiv comment11 pages