On the entangled ergodic theorem

Eisner, Tanja; Kunszenti-Kovács, Dávid

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 1, pp. 141-156.

We study the convergence of the so-called entangled ergodic averages

$$\frac{1}{N^k}\sum _{n_1,...,n_k=1}^N{T}_m^{n_{\alpha \left(m\right)}}{A}_{m-1}{T}_{m-1}^{n_{\alpha (m-1)}}{A}_{m-2}...A_1{T}_1^{n_{\alpha \left(1\right)}},$$

where $k\le m$ and $\alpha :\left\{1,...,m\right\}\to \left\{1,...,k\right\}$ is a surjective map. We show that, on general Banach spaces and without any restriction on the partition $\alpha$, the above averages converge strongly as $N\to \infty$ 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.

Classification : 47A35, 37A30

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