# On invariant measures for power bounded positive operators

Studia Mathematica (1996)

- Volume: 120, Issue: 2, page 183-189
- ISSN: 0039-3223

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topSato, Ryotaro. "On invariant measures for power bounded positive operators." Studia Mathematica 120.2 (1996): 183-189. <http://eudml.org/doc/216329>.

@article{Sato1996,

abstract = {We give a counterexample showing that $\overline\{(I-T*)L_\{∞\}\} ∩ L^\{+\}_\{∞\} = \{0\}$ does not imply the existence of a strictly positive function u in $L_1$ with Tu = u, where T is a power bounded positive linear operator on $L_1$ of a σ-finite measure space. This settles a conjecture by Brunel, Horowitz, and Lin.},

author = {Sato, Ryotaro},

journal = {Studia Mathematica},

keywords = {power bounded and Cesàro bounded positive operators; invariant measures; $L_1$ spaces; Cesàro bounded positive operators; spaces; strictly positive function; power bounded positive linear operator},

language = {eng},

number = {2},

pages = {183-189},

title = {On invariant measures for power bounded positive operators},

url = {http://eudml.org/doc/216329},

volume = {120},

year = {1996},

}

TY - JOUR

AU - Sato, Ryotaro

TI - On invariant measures for power bounded positive operators

JO - Studia Mathematica

PY - 1996

VL - 120

IS - 2

SP - 183

EP - 189

AB - We give a counterexample showing that $\overline{(I-T*)L_{∞}} ∩ L^{+}_{∞} = {0}$ does not imply the existence of a strictly positive function u in $L_1$ with Tu = u, where T is a power bounded positive linear operator on $L_1$ of a σ-finite measure space. This settles a conjecture by Brunel, Horowitz, and Lin.

LA - eng

KW - power bounded and Cesàro bounded positive operators; invariant measures; $L_1$ spaces; Cesàro bounded positive operators; spaces; strictly positive function; power bounded positive linear operator

UR - http://eudml.org/doc/216329

ER -

## References

top- [1] A. Brunel, Sur quelques problèmes de la théorie ergodique ponctuelle, Thèse, University of Paris, 1966.
- [2] A. Brunel, S. Horowitz and M. Lin, On subinvariant measures for positive operators in ${L}_{1}$, Ann. Inst. H. Poincaré Probab. Statist. 29 (1993), 105-117. Zbl0805.47030
- [3] Y. Derriennic and M. Lin, On invariant measures and ergodic theorems for positive operators, J. Funct. Anal. 13 (1973), 252-267. Zbl0262.28011
- [4] H. Fong, On invariant functions for positive operators, Colloq. Math. 22 (1970), 75-84. Zbl0223.28016
- [5] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985.
- [6] R. Sato, Ergodic properties of bounded ${L}_{1}$-operators, Proc. Amer. Math. Soc. 39 (1973), 540-546. Zbl0239.47003
- [7] L. Sucheston, On the ergodic theorem for positive operators I, Z. Wahrsch. Verw. Gebiete 8 (1967), 1-11. Zbl0175.05103

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