Disjointly homogeneous Banach lattices and compact products of operators
type: Item
instance of: scholarly article
title: Disjointly homogeneous Banach lattices and compact products of operators
zbMATH Open document ID: 1171.47016
author: Julio Flores
author: P. Tradacete
author: Vladimir G. Troitsky
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-05-19T00:00:00Z
review text: A Banach lattice \(E\) is called disjointly homogeneous if, for two arbitrary disjoint sequences in \(E\), there exist subsequences which are equivalent. Examples of disjointly homogeneous spaces include the spaces \(L_p(\mu)\) for \( 1\leq p \leq \infty \) and Lorentz spaces \(L_{p,q }(\mu)\). In general, a Banach lattice is called \(p\)-disjointly homogeneous if every normalized disjoint sequence has a subsequence equivalent to the unit vector basis of \(l^p\) \((c_0{\text{ if}}\,\, p = \infty \)). An interesting example is given to show that not every disjointly homogeneous Banach lattice is \(p\)-disjointly homogeneous. If \(E\) is a disjointly homogeneous Banach lattice, then either \(E\) or \(E'\) or both is a KB-space. If \(E\) is an \(\infty\)-disjointly homogeneous Banach lattice, then \(E'\) is a 1-disjointly homogeneous Banach lattice. Examples are provided to show that this duality is not true in general. In the second part of the paper, the authors study regular operators on disjointly homogeneous Banach lattices. The main result of this part is the following Theorem. Suppose that \(E\) is a disjointly homogeneous Banach lattice with order continuous norm and a weak unit. Let \(S,T\) be regular operators on \(E\) such that \(S\) is disjointly strictly singular and \(T\) is AM-compact. Then if \(E'\) has order continuous norm then \(ST\) is a compact operator. If \(E'\) does not have order continuous norm, then \(TS\) is a compact operator.
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 5555223
zbMATH Keywords: positive operator
zbMATH Keywords: disjointly strictly singular operator
zbMATH Keywords: AM-compact operator
zbMATH Keywords: Dunford-Pettis operator
reviewed by: Şafak Alpay
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.01.025
OpenAlex ID: W2079337854
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DOI: 10.1016/J.JMAA.2009.01.025
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description: scientific article; zbMATH DE number 5555223