Ein starkes ``Null-Zwei''-Gesetz in \(L^ p\). (A strong ``zero-two'' law in \(L^ p)\)
type: Item
instance of: scholarly article
title: Ein starkes ``Null-Zwei''-Gesetz in \(L^ p\). (A strong ``zero-two'' law in \(L^ p)\)
zbMATH Open document ID: 0641.46012
DOI: 10.1007/BF01215191
author: Rainer Wittman
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183728
review text: Let T be a positive linear contraction on the space \(L^ p=L^ p(\mu)\) (p\(\geq 1\) is fixed) with \(\sigma\)-finite measure \(\mu\). The following two theorems are the main results of the presented paper. Theorem 1.2. The inequality \(\inf_{n\in {\mathbb{N}}}\| | T^ n- T^{n+1}| \|_{L^ p}<2\) implies that \(\lim_{n\to \infty}\| | T^ n-T^{n+1}| \|_{L^ p}=0.\) Theorem 2.2 If there exist \(\epsilon >0\) such that \[ \inf_{n\in {\mathbb{N}}}\| | T^ n-T^{n+1}| f\|_{L^ p}\leq 2-\epsilon \quad (f\in L^ p) \] then \[ \lim_{n\to \infty}\| | T^ n- T^{n+1}| f\|_{L^ p}=0\quad (f\in L^ p). \]
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B60
zbMATH DE Number: 4044463
zbMATH Keywords: positive linear contraction
reviewed by: Alexander A. Mekler
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: Sets of multiplicity in locally compact abelian groups
cites work: The modulus of a regular linear operator and the ''zero-two'' law in \(L^ p\)-spaces \((1<p<+\infty,p\neq 2)\)
Recommended article: A Note on the L p Analogue of the "Zero-Two" Law
Recommended article: The strong ``zero-two'' law for positive contractions of Banach-Kantorovich \(L_p\)-lattices
Recommended article: An extension of the zero-two law on \(L_p\)-spaces
Recommended article: ON A MULTI-PARAMETRIC GENERALIZATION OF THE UNIFORM ZERO-TWO LAW IN L1-SPACES
Recommended article: Q3417799
Recommended article: Analogues of the ``zero-two`` law for positive linear contractions in L p and C(X)
Recommended article: The modulus of a regular linear operator and the ''zero-two'' law in \(L^ p\)-spaces \((1<p<+\infty,p\neq 2)\)
Recommended article: Q4404981
Recommended article: Uniform monotonicity of norms and the strong ``zero-two'' law
Recommended article: A strong comparison principle for the $p$-Laplacian
description: scientific article; zbMATH DE number 4044463