On ergodicity for operators with bounded resolvent in Banach spaces

Kirsti Mattila. "On ergodicity for operators with bounded resolvent in Banach spaces." Studia Mathematica 204.1 (2011): 63-72. <http://eudml.org/doc/285706>.

@article{KirstiMattila2011,
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 $||α(α - A)^\{-1\}||$ is uniformly bounded for all α > 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 $(T(t))_\{t≥0\}$ 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 δ, the operators T(t) - I, 0 < t < δ, are also ergodic.},
author = {Kirsti Mattila},
journal = {Studia Mathematica},
keywords = {ergodicity; bounded resolvent; canonical projection; semigroups of operators; Banach limits},
language = {eng},
number = {1},
pages = {63-72},
title = {On ergodicity for operators with bounded resolvent in Banach spaces},
url = {http://eudml.org/doc/285706},
volume = {204},
year = {2011},
}

TY - JOUR
AU - Kirsti Mattila
TI - On ergodicity for operators with bounded resolvent in Banach spaces
JO - Studia Mathematica
PY - 2011
VL - 204
IS - 1
SP - 63
EP - 72
AB - 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 $||α(α - A)^{-1}||$ is uniformly bounded for all α > 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 $(T(t))_{t≥0}$ 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 δ, the operators T(t) - I, 0 < t < δ, are also ergodic.
LA - eng
KW - ergodicity; bounded resolvent; canonical projection; semigroups of operators; Banach limits
UR - http://eudml.org/doc/285706
ER -