Positive compact operators on Banach lattices: Some loose ends
type: Item
instance of: scholarly article
title: Positive compact operators on Banach lattices: Some loose ends
zbMATH Open document ID: 0984.47032
DOI: 10.1023/A:1009838931430
author: R. Smith
published in: Positivity
publication date: 2002-05-14T00:00:00Z
review text: The following two topics are the main concern of this paper. Let \(0\leq S\leq T\) be two operators on a Banach lattice \(E\) and assume that \(T\) is compact. As the well known theorem of C. D. Aliprantis and O. Burkinshaw states the operator \(S^3\) is also compact, and if \(E\) or \(E^*\) has order continuous norm, then \(S^2\) is compact. Moreover, the well known theorem of P. Dodds and D. Fremlin asserts that if both \(E\) and \(E^*\) have order continuous norm, then \(S\) itself is compact. The author investigates as to what extent the order continuity condition in the above results is actually necessary. The second topic is aimed at studying the closure in the regular-norm topology of finite rank operators. For precise statements the reader is referred to this interesting paper.
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 47B06
zbMATH DE Number: 1538738
zbMATH Keywords: order continuous norm
zbMATH Keywords: order continuity condition
zbMATH Keywords: regular-norm topology
zbMATH Keywords: finite rank operators
reviewed by: Y. A. Abramovich
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1538738