Local \(\Delta _2^E(x)\) condition as a crucial tool for local structure of Calderón-Lozanovskiĭ spaces
type: Item
instance of: scholarly article
title: Local \(\Delta _2^E(x)\) condition as a crucial tool for local structure of Calderón-Lozanovskiĭ spaces
zbMATH Open document ID: 1211.46012
author: Ryszard Płuciennik
author: Paweł Kolwicz
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-06-18T00:00:00Z
review text: Given a \(\sigma\)-finite measure space \((T,\Sigma,\mu)\), a Köthe space \((E, \|\cdot\|_E)\) is a Banach space such that it is a subspace of the set of all \(\mu\)-measurable real valued functions \(L^0\) satisfying the conditions: (i) if \(x\in E\), \(y\in L^0\) and \(|y|\leq |x|\) a.e., then \(y\in E\) and \(\|y\|_E\leq \|x\|_E\); (ii) there exists a function \(x\in E\) positive on \(T\). Let \(\varphi : \mathbb{R} \rightarrow [0,\infty]\) be an Orlicz function such that it is convex, even, vanishing and continuous at zero, left continuous on \((0,\infty)\) and not identically equal to zero. Define on \(L^0\) a convex semimodular \(I_\varphi\) by \(I_\varphi(x) = \|\varphi\circ x\|_E\) if \(\varphi\circ x\in E\), and \(I_\varphi(x) = \infty\) otherwise. Then the Calderón-Lozanovskii space \(E_\varphi = \{x\in L^0: \exists{c>0} \;I_\varphi(cx) < \infty\}\) is a Banach space equipped with the norm \(\|x\|_\varphi = \inf\{\lambda>0: I_\varphi(x/\lambda) \leq 1\}\). A criterion for \(x\in E_\varphi\) is presented to be order continuous in terms of a new local growth condition \(\Delta_2(x)\). It is also proved that a Köthe sequence space \(E\) is lower locally uniformly monotone if and only if \(E\) is lower monotone and order continuous. The last result characterizes lower local uniform monotone points in \(E_\varphi\) in terms of \(\Delta_2(x)\) as well as some other useful parameters.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46B45
zbMATH DE Number: 5566259
zbMATH Keywords: Köthe spaces
zbMATH Keywords: Calderón-Lozanovskii spaces
zbMATH Keywords: order continuity
zbMATH Keywords: local monotonicity properties
reviewed by: Anna Kamińska
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.03.030
OpenAlex ID: W1967986010
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DOI: 10.1016/J.JMAA.2009.03.030
description: scientific article; zbMATH DE number 5566259