On power-subadditive positive operators on the $L_p$ spaces $(1<p<\infty )$
HTML articles powered by AMS MathViewer
- by Jean-Claude Lootgieter PDF
- Proc. Amer. Math. Soc. 145 (2017), 4299-4311 Request permission
Abstract:
In this article, we first give a counterexample to the statement appearing in a note of A. Brunel in 2002 concerning the study of positive operators on $L_p$ spaces $(1<p<\infty )$ whose sequence of powers is subadditive. We then give some properties of these operators when they are power-bounded.References
- M. A. Akcoglu, A pointwise ergodic theorem in $L_{p}$-spaces, Canadian J. Math. 27 (1975), no. 5, 1075–1082. MR 396901, DOI 10.4153/CJM-1975-112-7
- Antoine Brunel, Théorème ergodique pour les opérateurs positifs à moyennes bornées sur les espaces $L_p\;(1<p<\infty )$, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 195–207 (French, with English summary). MR 1176617, DOI 10.1017/S0143385700006684
- Antoine Brunel, Le théorème ergodique pour les opérateurs positifs sur les espaces $L_p\ (1<p<\infty )$ revisité, C. R. Math. Acad. Sci. Paris 334 (2002), no. 3, 205–207 (French, with English and French summaries). MR 1891059, DOI 10.1016/S1631-073X(02)02246-X
- Antoine Brunel and Richard Émilion, Sur les opérateurs positifs à moyennes bornées, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 6, 103–106 (French, with English summary). MR 741070, DOI 10.1017/S0143385700006684
- Yves Derriennic and Michael Lin, On invariant measures and ergodic theorems for positive operators, J. Functional Analysis 13 (1973), 252–267. MR 0355001, DOI 10.1016/0022-1236(73)90034-7
- R. Émilion, Mean-bounded operators and mean ergodic theorems, J. Funct. Anal. 61 (1985), no. 1, 1–14. MR 779737, DOI 10.1016/0022-1236(85)90037-0
- Nick Dungey, Subordinated discrete semigroups of operators, Trans. Amer. Math. Soc. 363 (2011), no. 4, 1721–1741. MR 2746662, DOI 10.1090/S0002-9947-2010-05094-9
- Markus Haase, The functional calculus for sectorial operators, Operator Theory: Advances and Applications, vol. 169, Birkhäuser Verlag, Basel, 2006. MR 2244037, DOI 10.1007/3-7643-7698-8
- A. Ionescu Tulcea, Ergodic properties of isometries in $L^{p}$ spaces, $1<p<\infty$, Bull. Amer. Math. Soc. 70 (1964), 366–371. MR 206207, DOI 10.1090/S0002-9904-1964-11099-5
- Charn Huen Kan, Ergodic properties of Lamperti operators, Canadian J. Math. 30 (1978), no. 6, 1206–1214. MR 511557, DOI 10.4153/CJM-1978-100-x
- Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411, DOI 10.1515/9783110844641
- Christian Le Merdy and Quanhua Xu, Strong $q$-variation inequalities for analytic semigroups, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, 2069–2097 (2013) (English, with English and French summaries). MR 3060752, DOI 10.5802/aif.2743
- F. J. Martín-Reyes and A. De la Torre, The dominated ergodic estimate for mean bounded, invertible, positive operators, Proc. Amer. Math. Soc. 104 (1988), no. 1, 69–75. MR 958045, DOI 10.1090/S0002-9939-1988-0958045-3
- R. Sato, On Brunel’s proof of a dominated ergodic theorem for positive linear operators on $L_p$ $(1< p<\infty )$ (quoted by A. Brunel in [3], p. 205).
- Alberto de la Torre, A simple proof of the maximal ergodic theorem, Canadian J. Math. 28 (1976), no. 5, 1073–1075. MR 417819, DOI 10.4153/CJM-1976-106-8
Additional Information
- Jean-Claude Lootgieter
- Affiliation: Université Pierre et Marie Curie, 3 place de l’Escadrille Normandie Niemen 75013, Paris, France
- MR Author ID: 239168
- Email: jean-claude.lootgieter@upmc.fr
- Received by editor(s): December 2, 2013
- Received by editor(s) in revised form: July 8, 2015, October 23, 2015, January 7, 2016, and January 21, 2016
- Published electronically: June 16, 2017
- Communicated by: Nimish A. Shaw
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 4299-4311
- MSC (2010): Primary 47A35
- DOI: https://doi.org/10.1090/proc/13124
- MathSciNet review: 3690614