On invariant measures for power bounded positive operators

Ryotaro Sato

Studia Mathematica (1996)

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

Abstract

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We give a counterexample showing that ( 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.

How to cite

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Sato, 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

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  1. [1] A. Brunel, Sur quelques problèmes de la théorie ergodique ponctuelle, Thèse, University of Paris, 1966. 
  2. [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. [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. [4] H. Fong, On invariant functions for positive operators, Colloq. Math. 22 (1970), 75-84. Zbl0223.28016
  5. [5] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985. 
  6. [6] R. Sato, Ergodic properties of bounded L 1 -operators, Proc. Amer. Math. Soc. 39 (1973), 540-546. Zbl0239.47003
  7. [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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