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Difference between revisions of "Zero-two law"

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A term used for a group of results dealing with the sequence $|T^{n+1} - T^n|$, where $T$ is a positive [[Contraction(2)|contraction]]. Usually, either this sequence converges to zero (uniformly or strongly), or for all $n$ the value two is associated with it. An example is the following result (see [[#References|[a1]]] and [[#References|[a2]]]). Let $T$ be a positive contraction on $L_p$, where $1 \le p \le \infty$. Then either $|\,\Vert T^{n+1} - T^n \Vert\,| = 2$ for all $n$ or $\Vert T^{n+1} - T^n \Vert \to 0$ as $n \to \infty$. For generalizations and additional references see [[#References|[a3]]].
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A term used for a group of results dealing with the sequence $|T^{n+1} - T^n|$, where $T$ is a positive [[Contraction(2)|contraction]]. Usually, either this sequence converges to zero (uniformly or strongly), or for all $n$ its norm is 2. An example is the following result (see [[#References|[a1]]] and [[#References|[a2]]]). Let $T$ be a positive contraction on $L_p$, where $1 \le p \le \infty$. Then either $\| | T^{n+1} - T^n | \| = 2$ for all $n$ or $\| | T^{n+1} - T^n | \| \to 0$ as $n \to \infty$. For generalizations and additional references see [[#References|[a3]]].
  
 
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Revision as of 21:38, 18 April 2017

A term used for a group of results dealing with the sequence $|T^{n+1} - T^n|$, where $T$ is a positive contraction. Usually, either this sequence converges to zero (uniformly or strongly), or for all $n$ its norm is 2. An example is the following result (see [a1] and [a2]). Let $T$ be a positive contraction on $L_p$, where $1 \le p \le \infty$. Then either $\| | T^{n+1} - T^n | \| = 2$ for all $n$ or $\| | T^{n+1} - T^n | \| \to 0$ as $n \to \infty$. For generalizations and additional references see [a3].

References

[a1] R. Zaharopol, "The modulus of a regular linear operator and the "zero-two" law in $L^p$-spaces ($1<p<\infty$, $p\ne2$)" J. Funct. Anal. , 68 (1986) pp. 300–312
[a2] Y. Katznelson, L. Tzafriri, "On power bounded operators" J. Funct. Anal. , 68 (1986) pp. 313–328
[a3] A.R. Schep, "A remark on the uniform zero-two law for positive contractions" Arch. Math. , 53 (1989) pp. 493–496
How to Cite This Entry:
Zero-two law. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zero-two_law&oldid=41114
This article was adapted from an original article by A.R. Schep (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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