View source for Zero-two law

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]]].

====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>