# On the mean ergodic theorem for Cesàro bounded operators

Colloquium Mathematicae (2000)

- Volume: 84/85, Issue: 2, page 443-455
- ISSN: 0010-1354

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topDerriennic, Yves. "On the mean ergodic theorem for Cesàro bounded operators." Colloquium Mathematicae 84/85.2 (2000): 443-455. <http://eudml.org/doc/210825>.

@article{Derriennic2000,

abstract = {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).},

author = {Derriennic, Yves},

journal = {Colloquium Mathematicae},

keywords = {Cesàro bounded operator; reflexive Banach space; ergodic theorem},

language = {eng},

number = {2},

pages = {443-455},

title = {On the mean ergodic theorem for Cesàro bounded operators},

url = {http://eudml.org/doc/210825},

volume = {84/85},

year = {2000},

}

TY - JOUR

AU - Derriennic, Yves

TI - On the mean ergodic theorem for Cesàro bounded operators

JO - Colloquium Mathematicae

PY - 2000

VL - 84/85

IS - 2

SP - 443

EP - 455

AB - 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).

LA - eng

KW - Cesàro bounded operator; reflexive Banach space; ergodic theorem

UR - http://eudml.org/doc/210825

ER -

## References

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- [H] G. H. Hardy, Divergent Series, Clarendon Press, 1949.
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- [Z] A. Zygmund, Trigonometric Series, Vol. 1, 2nd ed., Cambridge Univ. Press, 1959. Zbl0085.05601

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