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A note on the $ L\sp p$ analogue of the ``zero-two'' law

Author: Daniel Berend
Journal: Proc. Amer. Math. Soc. 114 (1992), 95-97
MSC: Primary 47A35; Secondary 47B38
MathSciNet review: 1057949
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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\Vert f ... ...eratorname{lim} _{n \to \infty }}\left\Vert {{T^n}f - {T^{n + 1}}f} \right\Vert$ 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$.

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  • [1] D. Ornstein and L. Sucheston, An operator theorem on $ {L^1}$ convergence to zero with applications to Markov kernels, Ann. Math. Statist. 41 (1970), 1631-1639. MR 0272057 (42:6938)
  • [2] R. Wittmann, Analogues of the "zero-two" law for positive linear contractions in $ {L^p}$ and $ C(X)$, Israel J. Math. 59 (1987), 8-28. MR 923659 (89d:47016)

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Keywords: "Zero-two" law, linear contraction, positive contraction
Article copyright: © Copyright 1992 American Mathematical Society

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