On Fixed Points and Multiparameter Ergodic Theorems in Banach Lattices

Millet, Annie ; Sucheston, Louis

Canadian journal of mathematics, Tome 40 (1988) no. 2, pp. 429-458

Résumé

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.

Détail

DOI : 10.4153/CJM-1988-017-5

Bibliographie

[1] 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] 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] 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] 4. Akcoglu, M. A. and Sucheston, L., An ergodic theorem on Banach lattices, Israël J. Math. 51 (1985), 208–222. Google Scholar

[5] 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] 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] 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] 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] 9. Chacon, R. V. and Ornstein, D. S., A general ergodic theorem, Illinois J. Math. 4 (1960), 153–160. Google Scholar

[10] 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] 11. Dowker, Y.N., On measurable transformations infinite measure spaces, Ann. Math. 62 (1955), 504–516. Google Scholar

[12] 12. Dunford, N., and Schwartz, J., Linear operators (Interscience Publ., (1958). Google Scholar

[13] 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] 14. Fong, H., On invariant functions for positive operators, Coll. Math. 22 (1970), 72–84. Google Scholar

[15] 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] 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] 17. Garsia, A., Topics in almost everywhere convergence, Lectures in advanced mathematics, Markham, (1970). Google Scholar

[18] 18. Granirer, E. E., On finite equivalent invariant measures for semigroups of transformations, Duke Math. J. 38 (1971), 395–408. Google Scholar

[19] 19. Tulcea, A. and C. Ionescu., Abstract ergodic theorems, Trans. Amer. Math. Soc. 187 (1963), 107–124. Google Scholar

[20] 20. Kakutani, S., Ergodic theory, Proc. International Congress of Mathematicians. 2 (1950), 129–142. Google Scholar

[21] 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] 22. Krengel, U., Ergodic theorems, De Gruyter Studies in Mathematics 6 (1985). Google Scholar

[23] 23. Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces II, function spaces (Springer, 1979). Google Scholar | DOI

[24] 24. Millet, A. and Sucheston, L., Demiconvergence of processes indexed by two indices, Ann. Inst. Henri Poincaré 19 (1983), 175–187. Google Scholar

[25] 25. Neveu, J., Bases mathématiques du calcul des probabilités (Masson, 1970). Google Scholar

[26] 26. Schaefer, H. H., Banach lattices and positive operators (Springer, 1974). Google Scholar | DOI

[27] 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] 28. Sucheston, L., On one-parameter proofs of almost sure convergence of multiparameter processes, Z. Wahrscheinlichkeitstheorie verw. Geb. 63 (1984), 43–49. Google Scholar