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On Fixed Points and Multiparameter Ergodic Theorems in Banach Lattices

Published online by Cambridge University Press:  20 November 2018

Annie Millet  and
Louis Sucheston
Annie Millet
Affiliation:
Université d'Angers, Angers, France
Louis Sucheston
Affiliation:
The Ohio State University, Columbus, Ohio
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We present here multiparameter results about positive operators acting on a weakly sequentially complete Banach lattice. Sections 1, 2 and 3 generalize results obtained by M. A. Akcoglu and the second author in the case of a contraction. Even in that case, the classical L1 theory extends to Banach lattices only under an additional monotonicity assumption (C), introduced in [3], without which the TL (or stochastic) ergodic theorem fails. The example proving this in [4] also shows that, without (C), the decomposition of the space into the “positive” part P, the largest support of a T-invariant element, and the “null” part N on which the TL limit is zero (see, e.g., [22], p. 141), also fails. If T is not a contraction but only mean-bounded, then the space decomposes into the “remaining” part Y, the largest support of a T*-invariant element, and the “disappearing part“ Z (see, e.g., [22], p. 172). Here we obtain, for Banach lattices and in the multiparameter case, a unified proof of both decompositions, and of the TL ergodic theorem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 2 , 01 April 1988 , pp. 429 - 458
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Akcoglu, M. A. and Sucheston, L., A ratio ergodic theorem for super additive processes, Z. Wahrscheinlichkeitstheorie verw. Geb. 44 (1978), 269–278.Google Scholar
2. Akcoglu, M. A. and Sucheston, L., A stochastic ergodic theorem for superadditive processes, Ergodic Theory and Dynamical Systems 3 (1983), 335–344.Google Scholar
3. Akcoglu, M. A. and Sucheston, L., On ergodic theory and truncated limits in Banach lattices, Proceedings of the 1983 Oberwolfach Measure Theory Conference, Lecture Notes in Math. 1089 (Springer-Verlag, Berlin, (1984), 241–262.Google Scholar
4. Akcoglu, M. A. and Sucheston, L., An ergodic theorem on Banach lattices, Israël J. Math. 51 (1985), 208–222.Google Scholar
5. Akcoglu, M. A. and Sucheston, L., On uniform ergodicity of norms and ergodic theorems in function spaces, Supplemento ai Rendiconti del Circolo Matematico di Palermo 8 (1985), 325–335.Google Scholar
6. Brunei, A., Théorème ergodique ponctuel pour un semigroup commutatif finiment engendré de contractions de L1 , Ann. Inst. Henri Poincaré 9 (1973), 327–343.Google Scholar
7. Brunei, A. and Emilion, R., Sur les opérateurs positifs à moyennes bornées, C.R. Acad. Se. Paris 298 (1984), 103–106.Google Scholar
8. Brunei, A. and Sucheston, L., Sur l'existence d'éléments invariants dans le treillis de Banach, C.R. Acad. Se. Paris 300 (1985), 59–63.Google Scholar
9. Chacon, R. V. and Ornstein, D. S., A general ergodic theorem, Illinois J. Math. 4 (1960), 153–160.Google Scholar
10. Derriennic, Y. and Lin, M., On invariant measures and ergodic theorems for positive operators, J. Functional Analysis 13 (1973), 252–267.Google Scholar
11. Dowker, Y.N., On measurable transformations infinite measure spaces, Ann. Math. 62 (1955), 504–516.Google Scholar
12. Dunford, N., and Schwartz, J., Linear operators (Interscience Publ., (1958).Google Scholar
13. Edgar, G. A. and Sucheston, L., Démonstrations de lois des grands nombres par les sous-martingales descendantes, C.A. Acad. Se. Paris 292 (1981), 967–969.Google Scholar
14. Fong, H., On invariant functions for positive operators, Coll. Math. 22 (1970), 72–84.Google Scholar
15. Frangos, N., and Sucheston, L., On convergence and demiconvergence of block martingales and submartingales, Proceedings of the Fifth International Conference on Probability in Banach Spaces, Tufts University, 1984. Lecture Notes in Math. 1153 (1985), 198–225.Google Scholar
16. Frangos, N., and Sucheston, L., On multiparameter ergodic and martingale theorems in infinite measure spaces, Probab. Th. Rel. Fields 71 (1986), 477–490.Google Scholar
17. Garsia, A., Topics in almost everywhere convergence, Lectures in advanced mathematics, Markham, (1970).Google Scholar
18. Granirer, E. E., On finite equivalent invariant measures for semigroups of transformations, Duke Math. J. 38 (1971), 395–408.Google Scholar
19. Tulcea, A. and C. Ionescu., Abstract ergodic theorems, Trans. Amer. Math. Soc. 187 (1963), 107–124.Google Scholar
20. Kakutani, S., Ergodic theory, Proc. International Congress of Mathematicians. 2 (1950), 129–142.Google Scholar
21. Krengel, U., On the global limit behavior of Markov chains and general non-singular Markov processes, Z. Wahrscheinlichkeitstheorie verw. Gebiete 6 (1966), 302–316.Google Scholar
22. Krengel, U., Ergodic theorems, De Gruyter Studies in Mathematics 6 (1985).Google Scholar
23. Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces II, function spaces (Springer, 1979).CrossRefGoogle Scholar
24. Millet, A. and Sucheston, L., Demiconvergence of processes indexed by two indices, Ann. Inst. Henri Poincaré 19 (1983), 175–187.Google Scholar
25. Neveu, J., Bases mathématiques du calcul des probabilités (Masson, 1970).Google Scholar
26. Schaefer, H. H., Banach lattices and positive operators (Springer, 1974).CrossRefGoogle Scholar
27. Sucheston, L., On the ergodic theorem for positive operators I, II, Z. Wahrscheinlichkeitstheorie verw. Geb. 8 (1967), 1–11, 353–356.Google Scholar
28. Sucheston, L., On one-parameter proofs of almost sure convergence of multiparameter processes, Z. Wahrscheinlichkeitstheorie verw. Geb. 63 (1984), 43–49.Google Scholar