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. 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