A note on the $L^ p$ analogue of the “zero-two” law
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- by Daniel Berend
- Proc. Amer. Math. Soc. 114 (1992), 95-97
- DOI: https://doi.org/10.1090/S0002-9939-1992-1057949-4
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Abstract:
It was proved by R. Wittmann [2] that, given a positive linear contraction of ${L^p}\left ( {1 \leq p < \infty } \right ),{\operatorname {sup} _{{{\left \| f \right \|}_p} \leq 1}}{\operatorname {lim} _{n \to \infty }}\left \| {{T^n}f - {T^{n + 1}}f} \right \|$ is either $\geq {\alpha _p}{\text {or}}0$; the (best possible) value of ${\alpha _p}$ is the ${l_p}$-norm of a certain $3 \times 3$ matrix. In this paper ${\alpha _p}$ is explicitly expressed as a function of $p$.References
- Donald Ornstein and Louis Sucheston, An operator theorem on $L_{1}$ convergence to zero with applications to Markov kernels, Ann. Math. Statist. 41 (1970), 1631–1639. MR 272057, DOI 10.1214/aoms/1177696806
- Rainer 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. MR 923659, DOI 10.1007/BF02779664
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 114 (1992), 95-97
- MSC: Primary 47A35; Secondary 47B38
- DOI: https://doi.org/10.1090/S0002-9939-1992-1057949-4
- MathSciNet review: 1057949