Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T04:32:54.157Z Has data issue: false hasContentIssue false
  • English
  • Français

Semi-Groups in L∞ And Local Ergodic Theorem

Published online by Cambridge University Press:  20 November 2018

R. Emilion
R. Emilion*
Affiliation:
Université Paris Vi Laboratoire de Probabilités 4, Place Jussieu 75230, Paris Cedex 05
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that any W*-continuous semi-group in L∞ is L1-norm continuous. As an application we prove the n-dimensional local ergodic theorem in L∞. We also note that any bounded additive process in L∞ is absolutely continuous.

For n = 1 this local theorem improves those of R. Sato [14] and D. Feyel [6] and for n ≥ 1 it generalizes M. Lin's ones which hold for positive operators [12].

Keywords

47A3547D0528A65
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 2 , 01 June 1986 , pp. 146 - 153
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Akcoglu, M.A. and A. del Junco, Differentiation of additive processes, Canad. J. Math. 33, (1981) pp. 749–768 .Google Scholar
2. Akcoglu, M.A. and Krengel, U., A differentiation theorem for additive processes. Math. Z. 163 (1978), pp. 199–210.Google Scholar
3. Akcoglu, M.A. and Krengel, U., A differentiation theorem in Lp. Math. Z. 169 (1979), pp. 31–40.Google Scholar
4. Emilion, R., Continuity at 0 of semi-groups in Lt and differentiation of additive processes, Ann. Inst. H. Poincaré. (to appear).Google Scholar
5. Emilion, R., Additive and superadditive local theorems, (to appear).Google Scholar
6. Feyel, D., Sur une classe remarquable de processus abeliens. Math. Z. (to appear).Google Scholar
7. Hille, E. and Phillips, R.S., Functional analysis and semi-groups, (A.M.S. colloquium publications, Providence, 1957).Google Scholar
8. Kipnis, C., Majoration des semi-groupe s de contractions de Lt et applications, Ann. Inst. Poincaré Sect. B (N.S.), 10 (1974), pp. 369–384.Google Scholar
9. Krengel, U., A necessary and sufficient condition for the validity of the local ergodic theorem, Lecture Notes in Math. n° 89, 170-177; (Springer, Berlin - Heidelberg - New York, 1969).Google Scholar
10. Krengel, U., A local ergodic theorem, Invent. Math. 6 (1969), pp. 329–333.Google Scholar
11. Lin, M., Semi-groups of Markov operators, Boll. Un. Mat. Ital., (4) 6 (1972), pp. 20–44.Google Scholar
12. Lin, M., On local ergodic convergence of semi-group s and additive processes. Israel J. Math. Vol. 42, (1982), pp. 300–308.Google Scholar
13. Sato, R., On local ergodic theorems for positive semi-groups, Studia Math., 63, (1978), pp. 45—55.Google Scholar
14. Sato, R., Two local ergodic theorems on L», J. Math. Soc. Japan, Vol. 32, n° 3, 1980, pp. 415–423.Google Scholar
15. Wiener, N., The ergodic theorem, Duke Math. J. 5 (1939), pp. 1–18.Google Scholar