On invariant measures for power bounded positive operators

• Autorzy: Ryotaro Sato
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

power bounded and Cesàro bounded positive operators; invariant measures; $L_1$ spaces

Kategorie tematyczne

• 47A35: Ergodic theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 120
• Numer: 2
• Strony: 183-189

Daty

wydano

1996

otrzymano

1996-01-29

poprawiono

1996-04-22

Twórcy

autor

Ryotaro Sato

• Department of Mathematics, Faculty of Science, Okayama University, Okayama, 700 Japan

Bibliografia

• [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.
• [3] Y. Derriennic and M. Lin, On invariant measures and ergodic theorems for positive operators, J. Funct. Anal. 13 (1973), 252-267.
• [4] H. Fong, On invariant functions for positive operators, Colloq. Math. 22 (1970), 75-84.
• [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.
• [7] L. Sucheston, On the ergodic theorem for positive operators I, Z. Wahrsch. Verw. Gebiete 8 (1967), 1-11.