We give a counterexample showing that does not imply the existence of a strictly positive function u in with Tu = u, where T is a power bounded positive linear operator on of a σ-finite measure space. This settles a conjecture by Brunel, Horowitz, and Lin.
@article{bwmeta1.element.bwnjournal-article-smv120i2p183bwm,
author = {Ryotaro Sato},
title = {On invariant measures for power bounded positive operators},
journal = {Studia Mathematica},
volume = {119},
year = {1996},
pages = {183-189},
zbl = {0861.47007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv120i2p183bwm}
}
Sato, Ryotaro. On invariant measures for power bounded positive operators. Studia Mathematica, Tome 119 (1996) pp. 183-189. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv120i2p183bwm/
[00000] [1] A. Brunel, Sur quelques problèmes de la théorie ergodique ponctuelle, Thèse, University of Paris, 1966.
[00001] [2] A. Brunel, S. Horowitz and M. Lin, On subinvariant measures for positive operators in , Ann. Inst. H. Poincaré Probab. Statist. 29 (1993), 105-117. | Zbl 0805.47030
[00002] [3] Y. Derriennic and M. Lin, On invariant measures and ergodic theorems for positive operators, J. Funct. Anal. 13 (1973), 252-267. | Zbl 0262.28011
[00003] [4] H. Fong, On invariant functions for positive operators, Colloq. Math. 22 (1970), 75-84. | Zbl 0223.28016
[00004] [5] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985.
[00005] [6] R. Sato, Ergodic properties of bounded -operators, Proc. Amer. Math. Soc. 39 (1973), 540-546. | Zbl 0239.47003
[00006] [7] L. Sucheston, On the ergodic theorem for positive operators I, Z. Wahrsch. Verw. Gebiete 8 (1967), 1-11. | Zbl 0175.05103