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## Studia Mathematica

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## On ergodicity for operators with bounded resolvent in Banach spaces

### Tom 204 / 2011

Studia Mathematica 204 (2011), 63-72 MSC: Primary 47A35, 47D06, 46B20. DOI: 10.4064/sm204-1-4

#### Streszczenie

We prove results on ergodicity, i.e. on the property that the space is a direct sum of the kernel of an operator and the closure of its range, for closed linear operators $A$ such that $\| \alpha (\alpha - A)^{-1}\|$ is uniformly bounded for all $\alpha > 0$. We consider operators on Banach spaces which have the property that the space is complemented in its second dual space by a projection $P$. Results on ergodicity are obtained under a norm condition $\| I-2P\|\, \| I-Q\| < 2$ where $Q$ is a projection depending on the operator $A$. For the space of James we show that $\| I-2P\| <2$ where $P$ is the canonical projection of the predual of the space. If $(T(t))_{t\geq 0}$ is a bounded strongly continuous and eventually norm continuous semigroup on a Banach space, we show that if the generator of the semigroup is ergodic, then, for some positive number $\delta$, the operators $T(t)-I, \, 0 < t < \delta$, are also ergodic.