On the mean ergodic theorem for Cesàro bounded operators
Colloquium Mathematicum, Tome 84 (2000) no. 2, pp. 443-455
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
For a Cesàro bounded operator in a Hilbert space or a reflexive Banach space the mean ergodic theorem does not hold in general. We give an additional geometrical assumption which is sufficient to imply the validity of that theorem. Our result yields the mean ergodic theorem for positive Cesàro bounded operators in $L^{p}$ (1 p ∞). We do not use the tauberian theorem of Hardy and Littlewood, which was the main tool of previous authors. Some new examples, interesting for summability theory, are described: we build an example of a mean ergodic operator T in a Hilbert space such that $∥T^{n}∥/n$ does not converge to 0, and whose adjoint operator is not mean ergodic (its Cesàro averages converge only weakly).
@article{10_4064_cm_84_85_2_443_455,
author = {Yves Derriennic},
title = {On the mean ergodic theorem for {Ces\`aro} bounded operators},
journal = {Colloquium Mathematicum},
pages = {443--455},
publisher = {mathdoc},
volume = {84},
number = {2},
year = {2000},
doi = {10.4064/cm-84/85-2-443-455},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm-84/85-2-443-455/}
}
TY - JOUR AU - Yves Derriennic TI - On the mean ergodic theorem for Cesàro bounded operators JO - Colloquium Mathematicum PY - 2000 SP - 443 EP - 455 VL - 84 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4064/cm-84/85-2-443-455/ DO - 10.4064/cm-84/85-2-443-455 LA - en ID - 10_4064_cm_84_85_2_443_455 ER -
Yves Derriennic. On the mean ergodic theorem for Cesàro bounded operators. Colloquium Mathematicum, Tome 84 (2000) no. 2, pp. 443-455. doi: 10.4064/cm-84/85-2-443-455
Cité par Sources :