The zero-two law for positive contractions is valid in all Banach lattices
type: Item
instance of: scholarly article
title: The zero-two law for positive contractions is valid in all Banach lattices
zbMATH Open document ID: 0657.47041
DOI: 10.1007/BF02787265
author: Helmut H. Schaefer
published in: Israel Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The author uses techniques from the theory of Banach lattices and positive operators in order to extend the so-called zero-two law [see e.g. \textit{Y. Katznelson} and \textit{L. Tzafriri}, J. Funct. Anal. 68, 313- 328 (1986; Zbl 0611.47005)] to arbitrary Banach lattices. More precisely, let T be a positive contraction on some complex Banach lattice X. Then either \(\lim_{n\to \infty}\| T^ n-T^{n+1}\| =0\) or \(\| T^ n-T^{n+1}\|_ r=2\) for all \(n\in {\mathbb{N}}\) whre \(\| \cdot \|_ r\) denotes the ``regular'' norm \(\| R\|_ r:=\inf \{\| S_ 1+S_ 2\|:R=S_ 1-S_ 2\), \(0\leq S_ i\in {\mathcal L}(X)\}\).
Mathematics Subject Classification ID: 47B60
zbMATH DE Number: 4073801
zbMATH Keywords: regular norm
zbMATH Keywords: Banach lattices and positive operators
zbMATH Keywords: zero-two law
zbMATH Keywords: positive contraction
Wikidata QID: Q114852587
MaRDI profile type: Publication
cites work: More on the "Zero-Two" Law
cites work: On power bounded operators
cites work: An Operator Theorem on $L_1$ Convergence to Zero with Applications to Markov Kernels
cites work: Q4049667
cites work: The modulus of a regular linear operator and the ''zero-two'' law in \(L^ p\)-spaces \((1<p<+\infty,p\neq 2)\)
description: scientific article; zbMATH DE number 4073801