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## A generalization of the uniform ergodic theorem to poles of arbitrary order

### Tom 122 / 1997

Studia Mathematica 122 (1997), 75-98 DOI: 10.4064/sm-122-1-75-98

#### Streszczenie

We obtain a generalization of the uniform ergodic theorem to the sequence $(1/n^{p}) ⅀^{n-1)_{k=0} T^k$, where T is a bounded linear operator on a Banach space and p is a positive integer. Indeed, we show that uniform convergence of the sequence above, together with an additional condition which is automatically satisfied for p = 1, is equivalent to 1 being a pole of the resolvent of T plus convergence to zero of $∥T^{n}∥/n^{p}$. Furthermore, we show that the two conditions above, together, are also equivalent to 1 being a pole of order less than or equal to p of the resolvent of T, plus a certain condition ℇ(k,p), which is less restrictive than convergence to zero of $∥T^{n}∥/n^{p}$ and generalizes the condition (called condition (ℇ-k)) introduced by K. B. Laursen and M. Mbekhta in their paper [LM2] (dealing with the case p=1).

#### Autorzy

• Laura Burlando

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