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 (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 does not converge to 0, and whose adjoint operator is not mean ergodic (its Cesàro averages converge only weakly).
@article{bwmeta1.element.bwnjournal-article-cmv84i2p443bwm,
author = {Yves Derriennic},
title = {On the mean ergodic theorem for Ces\`aro bounded operators},
journal = {Colloquium Mathematicae},
volume = {84/85},
year = {2000},
pages = {443-455},
zbl = {0968.47003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv84i2p443bwm}
}
Derriennic, Yves. On the mean ergodic theorem for Cesàro bounded operators. Colloquium Mathematicae, Tome 84/85 (2000) pp. 443-455. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv84i2p443bwm/
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