The problem of central orthomorphisms in a class of \(F\)-lattices
type: Item
instance of: scholarly article
title: The problem of central orthomorphisms in a class of \(F\)-lattices
zbMATH Open document ID: 1336.46006
author: M. Wójtowicz
author: Halina Wiśniewska
published in: Indagationes Mathematicae. New Series
publication date: 2015-03-23T00:00:00Z
review text: This paper is concerned with the following problem: \(( P)\) Characterize those Archimedean vector lattices \(E\) for which every order bounded orthomorphism on \(E\) is a central operator, that is, an operator satisfying \(|T | \leq \lambda I_E\) for some positive real number \(\lambda\), where \(I_E\) is the identity on \(E\). Let \(Z(E)\) denote the central operators on \(E\) and let \(\mathrm{Orth}(E)\) denote the orthomorphisms on \(E\). In general, we have \(Z(E) \subset \mathrm{Orth}(E)\). In [Compos. Math. 35, 225--238 (1977; Zbl 0381.47021)], \textit{A. W. Wickstead} showed that, if \(E\) is a Banach lattice, then \(Z(E) = \mathrm{Orth}(E)\). The introduction of the paper contains a brief history of the problem \((P)\). Recall that a subadditive function \(x \mapsto ||x||\) defined on a linear space \(X\) such that \(||x|| = 0\) if and only if \(x = 0\) and, for every \(x \in X\), the function \(t \mapsto ||tx||\) is continuous at zero, is called an \(F\)-norm on \(X\). If \(X\) is complete with respect to \(||\cdot||\), then \(X\) is called an \(F\)-space. An \(F\)-space \(X\) is called locally bounded at \(0\) if there exists an open ball in \(X\) centered at \(0\) which is topologically bounded. If the vector lattice \(E\) is endowed with a monotone \(F\)-norm \(||\cdot||\) under which \(E\) is complete, then \((E, ||\cdot||)\) is said to be an \(F\)-lattice. Every Banach lattice is an \(F\)-lattice. The main result of the paper is the following. Theorem. If \((E, ||\cdot||)\) is a locally bounded \(F\)-lattice, then \(\mathrm{Orth}(E) = Z(E)\). The preceding result is then applied to Nakano spaces. Let \((p_n )\) be a fixed sequence of numbers in the interval \((0, 1)\). Consider the space of sequences \(l^{(p_n )}\), where \[ l^{(p_n )}= \left\{(x_n ) :\sum^{\infty}_{n=1}|x_n |{}^{p_n} < \infty\right\}, \] endowed with the \(F\)-norm \(\displaystyle ||(x_n )|| =\sum^{\infty}_{ n=1} |x_n |{}^{p_n}\). The next result answers problem \((P)\) for the class…
reviewed by: Şafak Alpay
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46A45
zbMATH DE Number: 6418462
zbMATH Keywords: orthomorphism
zbMATH Keywords: \(F\)-lattice
zbMATH Keywords: locally bounded \(F\)-space
zbMATH Keywords: Musielak-Orlicz sequence space
MaRDI profile type: Publication
full work available at URL: j.indag.2014.12.003
OpenAlex ID: W1992939321
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DOI: 10.1016/J.INDAG.2014.12.003
description: scientific article; zbMATH DE number 6418462