On the mean ergodic theorem for Cesàro bounded operators

Yves Derriennic

Colloquium Mathematicae (2000)

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

Abstract

top
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).

How to cite

top

Derriennic, 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

top
  1. [B] H. Berliocchi, unpublished manuscript, 1983. 
  2. [DL] Y. Derriennic and M. Lin, On invariant measures and ergodic theorems for positive operators, J. Funct. Anal. 13 (1973), 252-267. Zbl0262.28011
  3. [DS] N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience, 1958. 
  4. [E1] R. Emilion, Opérateurs à moyennes bornées et théorèmes ergodiques en moyenne, C. R. Acad. Sci. Paris Sér. I 296 (1983), 641-643. Zbl0537.47004
  5. [E2] R. Emilion, Mean bounded operators and mean ergodic theorems, J. Funct. Anal. 61 (1985), 1-14. Zbl0562.47007
  6. [F] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, 1966. Zbl0138.10207
  7. [H] G. H. Hardy, Divergent Series, Clarendon Press, 1949. 
  8. [Hi] E. Hille, Remarks on ergodic theorems, Trans. Amer. Math. Soc. 57 (1945), 246-269. Zbl0063.02017
  9. [K] U. Krengel, Ergodic Theorems, de Gruyter, 1985. 
  10. [S] H. Schaefer, Banach Lattices and Positive Operators, Grundlehren Math. Wiss. 215, Springer, 1974. Zbl0296.47023
  11. [Z] A. Zygmund, Trigonometric Series, Vol. 1, 2nd ed., Cambridge Univ. Press, 1959. Zbl0085.05601

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.