ON A MULTI-PARAMETRIC GENERALIZATION OF THE UNIFORM ZERO-TWO LAW IN L1-SPACES

Title & Authors
ON A MULTI-PARAMETRIC GENERALIZATION OF THE UNIFORM ZERO-TWO LAW IN L1-SPACES
MUKHAMEDOV, FARRUKH;

Abstract
Following an idea of Ornstein and Sucheston, Foguel proved the so-called uniform "zero-two" law: let T : $\small{L^1}$(X, $\small{\mathcal{F}}$, $\small{{\mu}}$) $\small{{\rightarrow}}$ $\small{L^1}$(X, $\small{\mathcal{F}}$, $\small{{\mu}}$) be a positive contraction. If for some $\small{m{\in}{\mathbb{N}}{\cup}\{0\}}$ one has $\small{{\parallel}T^{m+1}-T^m{\parallel}}$ < 2, then $\lim_{n{\rightarrow}{\infty}}{\parallel}T^{m+1}-T^m{\parallel} Keywords multi parametric;positive contraction;"zero-two" law; Language English Cited by 1. On a genaralized uniform zero-two law for positive contractions of non-commutativeL1-spaces, Journal of Physics: Conference Series, 2016, 697, 012003 References 1. M. Akcoglu and J. Baxter, Tail field representations and the zero-two law, Israel J. Math. 123 (2001), 253-272. 2. C. D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, 2006. 3. Y. Derriennic, Lois "zero ou deux" pour les processes de Markov, Applications aux marches aleatoires, Ann. Inst. H. Poincare Sec. B 12 (1976), no. 2, 111-129. 4. S. R. Foguel, On the "zero-two" law, Israel J. Math. 10 (1971), 275-280. 5. S. R. Foguel, More on the "zero-two" law, Proc. Amer. Math. Soc. 61 (1976), no. 2, 262-264. 6. S. R. Foguel, A generalized 0-2 law, Israel J. Math. 45 (1983), no. 2-3, 219-224. 7. B. Jamison and S. Orey, Markov chains recurrent in the sense of Harris, Z. Wahrsch. Verw. Geb. 8 (1967), 41-48. 8. Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), no. 3, 313-328. 9. U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985. 10. M. Lin, On the "zero-two" law for conservative Markov operators, Z. Wahrsch. Verw. Geb. 61 (1982), no. 4, 513-525. 11. M. Lin, The uniform zero-two law for positive operators in Banach lattices, Studia Math. 131 (1998), no. 2, 149-153. 12. F. Mukhamedov, On dominant contractions and a generalization of the zero-two law, Positivity 15 (2011), no. 3, 497-508. 13. D. Orstein and L. Sucheston, An operator theorem on$L_1$convergence to zero with applications to Markov operators, Ann. Math. Statist. 41 (1970), 1631-1639. 14. H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, 1974. 15. H. H. Schaefer, The zero-two law for positive contractions is valid in all Banach lattices, Israel J. Math. 59 (1987), no. 2, 241-244. 16. A. Schep, A remark on the uniform zero-two law for positive contractions, Arch. Math. (Basel) 53 (1989), no. 5, 493-496. 17. R. Wittmann, Analogues of the "zero-two" law for positive linear contractions in$L_p$and C(X), Israel J. Math. 59 (1987), no. 1, 8-28. 18. R. Wittmann, Ein starkes "Null-Zwei"-Gesetz in$L_p$, Math. Z. 197 (1988), no. 2, 223-229. 19. R. Zaharopol, The modulus of a regular linear operator and the 'zero-two' law in$L^p$- spaces (1 < p < +${\infty}$, p 6${\not=}\$ 2), J. Funct. Anal. 68 (1986), no. 3, 300-312.

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