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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z1100201.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z1100202.png" /> is a positive [[Contraction(2)|contraction]]. Usually, either this sequence converges to zero (uniformly or strongly), or for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z1100203.png" /> the value two is associated with it. An example is the following result (see [[#References|[a1]]] and [[#References|[a2]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z1100204.png" /> be a positive contraction on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z1100205.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z1100206.png" />. Then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z1100207.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z1100208.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z1100209.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z11002010.png" />. 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. 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 $n \to \infty$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Zaharopol,  "The modulus of a regular linear operator and the  "zero-two"  law in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z11002011.png" />-spaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z11002012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z110/z110020/z11002013.png" />)"  ''J. Funct. Anal.'' , '''68'''  (1986)  pp. 300–312</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Y. Katznelson,  L. Tzafriri,  "On power bounded operators"  ''J. Funct. Anal.'' , '''68'''  (1986)  pp. 313–328</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.R. Schep,  "A remark on the uniform zero-two law for positive contractions"  ''Arch. Math.'' , '''53'''  (1989)  pp. 493–496</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  Y. Katznelson,  L. Tzafriri,  "On power bounded operators"  ''J. Funct. Anal.'' , '''68'''  (1986)  pp. 313–328</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  A.R. Schep,  "A remark on the uniform zero-two law for positive contractions"  ''Arch. Math.'' , '''53'''  (1989)  pp. 493–496</TD></TR>
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</table>
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Latest revision as of 21:41, 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. 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 $n \to \infty$.

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=15804
This article was adapted from an original article by A.R. Schep (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article