Analogues of the ``zero-two`` law for positive linear contractions in L p and C(X)
type: Item
instance of: scholarly article
title: Analogues of the ``zero-two`` law for positive linear contractions in L p and C(X)
zbMATH Open document ID: 0654.46034
DOI: 10.1007/BF02779664
author: Rainer Wittmann
published in: Israel Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let a \(\sigma\)-finite measure space be given, and choose a positive linear contraction (a nonexpansive mapping) T, defined on the related L p space \((1\leq p<\infty)\). The zero-two law is the statement that for \(p=1\) \[ (*)\quad \sup (\lim_{n\to \infty}\| T\quad nf-T^{n+1}f\|_ 1)\quad is\quad either\quad zero\quad or\quad two, \] the sup being taken over all f's in the unit ball of L 1, a result proved by Ornstein and Sucheston. In this paper, the author proves many variants of this theorem, including (i) if \(\alpha_ p\) is a `natural' constant which is 2 when \(p=1\), and is the best possible choice, then \[ \sup (\lim_{n\to \infty}\| T\quad nf-T^{n+1}f\|_ p\quad is\quad either\quad zero\quad or\quad \geq \alpha_ p, \] the sup being taken over the unit ball of L p; moreover, (ii) \[ \sup (\lim_{n\to \infty}\| T\quad nf- T^{n+1}f\|_ p\quad is\quad either\quad 2^{1/p}\quad or\quad zero, \] if the sup is taken over nonnegative f's in the unit ball of L p. There are also extensions concerning the uniform case of (*) (when sup and lim are interchanged, thus yielding operator norms), and the case of sup-norms (a.e. bounded or continuous functions); finally, some improvements are announced (1.10) and open questions raised (2.2).
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 60F25
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 4067613
zbMATH Keywords: positive operator
zbMATH Keywords: positive linear contraction
zbMATH Keywords: nonexpansive mapping
zbMATH Keywords: zero-two law
Wikidata QID: Q114852588
MaRDI profile type: Publication
cites work: Q4124070
cites work: More on the "Zero-Two" Law
cites work: La loi ''zéro ou deux'' et ses consequences pour le comportement asymptotique des opérateurs positifs
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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 4067613