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

### Volume 204 / 2011

#### Abstract

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.