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http://dx.doi.org/10.4134/BKMS.2015.52.6.1819

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

MUKHAMEDOV, FARRUKH (Department of Computational & Theoretical Sciences Faculty of Science, International Islamic University Malaysia)
Publication Information
Bulletin of the Korean Mathematical Society / v.52, no.6, 2015 , pp. 1819-1826 More about this Journal
Abstract
Following an idea of Ornstein and Sucheston, Foguel proved the so-called uniform "zero-two" law: let T : $L^1$(X, $\mathcal{F}$, ${\mu}$) ${\rightarrow}$ $L^1$(X, $\mathcal{F}$, ${\mu}$) be a positive contraction. If for some $m{\in}{\mathbb{N}}{\cup}\{0\}$ one has ${\parallel}T^{m+1}-T^m{\parallel}$ < 2, then $\lim_{n{\rightarrow}{\infty}}{\parallel}T^{m+1}-T^m{\parallel}=0$. There are many papers devoted to generalizations of this law. In the present paper we provide a multi-parametric generalization of the uniform zero-two law for $L^1$-contractions.
Keywords
multi parametric; positive contraction; "zero-two" law;
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  • Reference
1 M. Akcoglu and J. Baxter, Tail field representations and the zero-two law, Israel J. Math. 123 (2001), 253-272.   DOI
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.   DOI
5 S. R. Foguel, More on the "zero-two" law, Proc. Amer. Math. Soc. 61 (1976), no. 2, 262-264.   DOI
6 S. R. Foguel, A generalized 0-2 law, Israel J. Math. 45 (1983), no. 2-3, 219-224.   DOI
7 B. Jamison and S. Orey, Markov chains recurrent in the sense of Harris, Z. Wahrsch. Verw. Geb. 8 (1967), 41-48.   DOI
8 Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), no. 3, 313-328.   DOI
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.   DOI
11 M. Lin, The uniform zero-two law for positive operators in Banach lattices, Studia Math. 131 (1998), no. 2, 149-153.   DOI
12 F. Mukhamedov, On dominant contractions and a generalization of the zero-two law, Positivity 15 (2011), no. 3, 497-508.   DOI
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.   DOI
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.   DOI
16 A. Schep, A remark on the uniform zero-two law for positive contractions, Arch. Math. (Basel) 53 (1989), no. 5, 493-496.   DOI
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.   DOI
18 R. Wittmann, Ein starkes "Null-Zwei"-Gesetz in $L_p$, Math. Z. 197 (1988), no. 2, 223-229.   DOI
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.   DOI
20 R. Zaharopol, On the 'zero-two' law for positive contractions, Proc. Edinburgh Math. Soc. (2) 32 (1989), no. 3, 363-370.   DOI
21 R. Zaharopol, A local zero-two law and some applications, Turkish J. Math. 24 (2000), no. 1, 109-120.