View source for Zero-two law

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====
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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>
<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>
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