A remark on the uniform zero-two law for positive contractions
type: Item
instance of: scholarly article
title: A remark on the uniform zero-two law for positive contractions
zbMATH Open document ID: 0702.47019
DOI: 10.1007/BF01324724
author: Anton R. Schep
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: The author considers many recent zero-two law results for positive contractions on Banach lattices, i.e. results stating that if T is a positive contraction on a Banach lattice, then \(\| T^ n- T^{n+1}\| =2\) for all \(n\in {\mathbb{N}}\) or \(\| T^ n-T^{n+1}\| \to 0\) as \(n\to \infty\). Then he goes to show that this zero-two law holds for positive contractions on arbitrary Dedekind complete Banach lattices, this way improving some of the previous results. Furthermore, he is able to characterize T's such that \(\| T^ n(I-T^ k)\| \to 0\) as \(N\to \infty\) as exactly those T's for which the peripheral order spectrum contains no \(\lambda\) with \(| \lambda | =1\) and \(\lambda^ k\neq 1\). As a consequence one has that \(\| T^ n(I-T^ k)\| =2\) for all \(n\geq 0\), \(k\geq 1\) if and only if the peripheral order spectrum of T equals \(\{z:| z| =1\}\).
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 4151166
zbMATH Keywords: peripheral order spectrum
zbMATH Keywords: zero-two law results for positive contractions on Banach lattices
zbMATH Keywords: Dedekind complete Banach lattices
reviewed by: G. Emmanuele
MaRDI profile type: Publication
cites work: On the 0-spektrum of regular operators and the spectrum of measures
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cites work: Q5184011
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cites work: The zero-two law for positive contractions is valid in all Banach lattices
cites work: Analogues of the ``zero-two`` law for positive linear contractions in L p and C(X)
cites work: Ein starkes ``Null-Zwei''-Gesetz in \(L^ p\). (A strong ``zero-two'' law in \(L^ p)\)
cites work: Q3669960
cites work: The modulus of a regular linear operator and the ''zero-two'' law in \(L^ p\)-spaces \((1<p<+\infty,p\neq 2)\)
full work available at URL: https://doi.org/10.1007/bf01324724
OpenAlex ID: W2034860794
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description: scientific article; zbMATH DE number 4151166