Characterization of the order weak compactness of semi-compact operators
type: Item
instance of: scholarly article
title: Characterization of the order weak compactness of semi-compact operators
zbMATH Open document ID: 1190.47038
author: Belmesnaoui Aqzzouz
author: Aziz Elbour
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-06-10T00:00:00Z
review text: Recall that an operator \(T\) from a Banach lattice \(E\) to \(F\) is called semi-compact if for every \(\varepsilon>0\), there is a positive element \(u\) in \(F\) so that the image of the unit ball is contained in the order interval \([-u,u]\) plus \(\varepsilon\) times the unit ball in \(F\), and \(T\) is order weakly compact if for each positive \(x\) in \(E\), the image of the order interval \( [0,x]\) is relatively weakly compact. Relationships between these two types of operators are established. It is shown that for operators from \(E\) to \(F\) with \(F\) order \(\sigma\)-complete, the following are equivalent: (1) each order bounded operator is order weakly compact, (2) each semi-compact operator is order weakly compact, (3) each positive semi-compact operator is order weakly compact, (4) one of \(E\) and \(F\) has order continuous norm. For the same condition on \(F\), it is also established that if each positive semi-compact operator has has a semi-compact adjoint, then either \(E\) is a KB-space or \(F\) has an order continuous norm. These ideas are used to show that, without restriction on \(F\), if every positive operator is semi-compact if and only if its adjoint is semi-compact, then \(E'\) or \(F'\) has an order continuous norm. Examples and further results are provided.
reviewed by: William A. Feldman
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 5563813
zbMATH Keywords: semi-compact operator
zbMATH Keywords: order weakly compact operator
zbMATH Keywords: order continuous norm
zbMATH Keywords: KB-space
zbMATH Keywords: Banach lattice
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.01.072
OpenAlex ID: W1974414010
cites work: Q4746343
cites work: Semi-compactness of positive Dunford–Pettis operators on Banach lattices
cites work: The duality problem for the class of b-weakly compact operators
cites work: Banach lattices
cites work: Q4285293
cites work: Q3669960
DOI: 10.1016/J.JMAA.2009.01.072
description: scientific article; zbMATH DE number 5563813