The strong Gelfand-Phillips property in Banach lattices
type: Item
instance of: scholarly article
title: The strong Gelfand-Phillips property in Banach lattices
zbMATH Open document ID: 1335.46013
DOI: 10.1215/17358787-3158354
author: Halimeh Ardakani
author: S. M. Moshtaghioun
author: M. Salimi
author: Seyed Mohammad Sadegh Modarres Mosadegh
published in: Banach Journal of Mathematical Analysis
publication date: 2015-10-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bjma/1444913860
review text: Let \(X,Y\) be Banach spaces and \(E,F\) be Banach lattices. A subset \(A\subset X\) is called limited if every weak* null sequence \((x_n^*)\) in the dual \(X^*\) converges uniformly on \(A\). If every limited subset of \(X\) is relatively compact, then \(X\) is said to have Gelfand-Phillips property (GPP). \(c_0,l_1\), every separable Banach space, every Schur space has the GP property. A subset \(A\) of \(E\) is almost limited if every disjoint weak* null sequence \((x_n^*)\) in \(E^*\) converges uniformly on \(A\). \(E\) is said to have the strong Gelfand-Phillips property (SGP property) if every almost limited subset in \(E\) is relatively compact. The strong Gelfand-Phillips property implies the Gelfand-Phillips property. Discrete Banach lattices with order continuous norms have the SGP property and if, in addition, \(E\) is \(\sigma\)-Dedekind complete, then \(E\) has the SGP property if and only if \(E\) is discrete and has order continuous norm. If \(E\) is a nondiscrete Banach lattice with order continuous norm, then \(E\) does not have the SGP property. \(X\) has the GP property if and only if every weakly null and limited sequence in \(X\) is norm null. Similarly, if \(E\) has the SGP property, then every weakly null and almost limited sequence in \(E\) is norm null. The converse is false, in general. If \(E\) has order continuous norm and \(A\subset E\) is almost limited in \(E\), then not only is \(A\) conditionally weakly compact, but \(A\) is relatively weakly compact. Weakly null and almost limited sequences in a Banach lattice are not norm null, in general. The Rademacher functions in \(L^1[0,1]\) are weakly null and almost limited but not norm null. If \(E^*\) has a weak order unit or \(E\) has order continuous norm, then \(E\) has the SGP property if and only if every weakly null and almost limited sequence in \(E\) is norm null. A bounded linear operator \(T:E\rightarrow X\) is called almost limited completely continuous (alcc) if it maps weakly…
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 46B50
Mathematics Subject Classification ID: 47B65
zbMATH DE Number: 6495770
zbMATH Keywords: almost limited set
zbMATH Keywords: limited completely continuous operator
zbMATH Keywords: Gelfand-Phillips property
reviewed by: Şafak Alpay
MaRDI profile type: Publication
OpenAlex ID: W2126083334
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description: scientific article; zbMATH DE number 6495770