On ergodicity for operators with bounded resolvent in Banach spaces

Kirsti Mattila¹

¹ Department of Mathematics Royal Institute of Technology SE-10044 Stockholm, Sweden

Studia Mathematica, Tome 204 (2011) no. 1, pp. 63-72

Résumé

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.

Détail

DOI : 10.4064/sm204-1-4

Keywords: prove results ergodicity property space direct sum kernel operator closure its range closed linear operators alpha alpha uniformly bounded alpha consider operators banach spaces which have property space complemented its second dual space projection results ergodicity obtained under norm condition i i q where projection depending operator space james i where canonical projection predual space geq bounded strongly continuous eventually norm continuous semigroup banach space generator semigroup ergodic positive number delta operators i delta ergodic