Compactness of L-weakly and M-weakly compact operators on Banach lattices
type: Item
instance of: scholarly article
title: Compactness of L-weakly and M-weakly compact operators on Banach lattices
zbMATH Open document ID: 1244.47033
author: Belmesnaoui Aqzzouz
author: Aziz Elbour
author: A. W. Wickstead
publication date: 2012-02-03T00:00:00Z
full work available at URL: LMWC.pdf
review text: Let \(E\) be a Banach lattice and \(X\) a Banach space. An operator \(T:E\rightarrow X\) is called M-weakly compact if \(\lim_n|| Tx_n|| = 0\) for each norm bounded disjoint sequence \((x_n)\) in \(E\). \(T\) is called L-weakly compact if \(\lim_n ||y_n|| = 0\) for each disjoint sequence \((y_n)\) in the solid hull of \(T(B_X)\), where \(B_X\) is the closed unit ball of \(X\). L- or M-weakly compact operators need not be compact. Similarly, compact operators need not be L- or M-weakly compact. \textit{Z. L. Chen} and \textit{A. W. Wickstead} [Indag. Math., New Ser. 10, No. 3, 321--336 (1999; Zbl 1028.47028)] have shown that M-weakly compact operators from an AL-space into a Banach space are compact and each L-weakly compact operator from a Banach space into an AM-space is compact. In this paper, these two results are generalized and the converse problem is studied. One of the main result is the following. Theorem. Assume that the dual \(E'\) of \(E\) does not have order continuous norm. Then the following are equivalent: (1) Every L-weakly compact operator \(T:E\rightarrow F\) is compact. (2) Each positive L-weakly compact operator \(T:E\rightarrow F\) is compact. (3) The largest lattice ideal in \(E\) for which the norm is order continuous is discrete. It is shown that, if \(E'\) is discrete with order continuous norm, then each regular L-weakly compact operator \(T:E\rightarrow F\) is compact and, in Theorem 3.6, this property is shown to give a characterization of Banach lattices whose duals are discrete and have order continuous norm. If the range \(Y\) contains an isomorphic copy of \(c_0\), then every M-weakly compact operator \(T:E\rightarrow Y\) is compact if and only if the largest lattice ideal \((E')^a\) in \(E'\) is discrete result. As with L-weakly compact operators, the authors prove the following on M-weakly compact operators. Theorem. The following are equivalent: (1) \(F\) is discrete and has order continuous norm. (2) For each \(E\), every regular …
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46B40
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 6003013
zbMATH Keywords: compact operator
zbMATH Keywords: L-weakly compact operator
zbMATH Keywords: M-weakly compact operator
zbMATH Keywords: order continuous norm
zbMATH Keywords: discrete vector lattice
reviewed by: Şafak Alpay
MaRDI profile type: Publication
OpenAlex ID: W1971382396
cites work: Q5491574
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cites work: Banach lattices
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cites work: Q4259426
cites work: Q3669960
DOI: 10.1007/S12215-011-0025-7
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 6003013