Vector lattices of weakly compact operators

on Banach lattices

Authors:
Z. L. Chen and A. W. Wickstead

Journal:
Trans. Amer. Math. Soc. **352** (2000), 397-412

MSC (1991):
Primary 47B65; Secondary 47B07

Published electronically:
July 21, 1999

MathSciNet review:
1641095

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Abstract: A result of Aliprantis and Burkinshaw shows that weakly compact operators from an AL-space into a KB-space have a weakly compact modulus. Groenewegen characterised the largest class of range spaces for which this remains true whenever the domain is an AL-space and Schmidt proved a dual result. Both of these authors used vector-valued integration in their proofs. We give elementary proofs of both results and also characterise the largest class of domains for which the conclusion remains true whenever the range space is a KB-space. We conclude by studying the order structure of spaces of weakly compact operators between Banach lattices to prove results analogous to earlier results of one of the authors for spaces of compact operators.

**[1]**Ju. A. Abramovič,*Weakly compact sets in topological 𝐾-spaces*, Teor. Funkciĭ Funkcional. Anal. i Priložen.**15**(1972), 27–35 (Russian). MR**0306860****[2]**Y. A. Abramovič, E. D. Positsel′skiĭ, and L. P. Yanovskiĭ,*On some parameters associated with normed lattices and on series characterisation of 𝑀-spaces*, Studia Math.**63**(1978), no. 1, 1–8. MR**508877****[3]**Yuri A. Abramovich,*When each continuous operator is regular*, Functional analysis, optimization, and mathematical economics, Oxford Univ. Press, New York, 1990, pp. 133–140. MR**1082571****[4]**Y. A. Abramovich and A. W. Wickstead,*A compact regular operator without modulus*, Proc. Amer. Math. Soc.**116**(1992), no. 3, 721–726. MR**1098395**, 10.1090/S0002-9939-1992-1098395-7**[5]**Y. A. Abramovich and A. W. Wickstead,*Recent results on the order structure of compact operators*, Irish Math. Soc. Bull.**32**(1994), 32–45. MR**1286401****[6]**Y. A. Abramovich and A. W. Wickstead,*Solutions of several problems in the theory of compact positive operators*, Proc. Amer. Math. Soc.**123**(1995), no. 10, 3021–3026. MR**1283534**, 10.1090/S0002-9939-1995-1283534-8**[7]**Y. A. Abramovich, Z. L. Chen and A.W. Wickstead,*Regular-norm balls can be closed in the strong operator topology*, Positivity**1**(1997), 75-96.**[8]**C. D. Aliprantis and O. Burkinshaw,*On weakly compact operators on Banach lattices*, Proc. Amer. Math. Soc.**83**(1981), no. 3, 573–578. MR**627695**, 10.1090/S0002-9939-1981-0627695-X**[9]**Charalambos D. Aliprantis and Owen Burkinshaw,*Positive operators*, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR**809372****[10]**Z. L. Chen and A. W. Wickstead,*Relative weak compactness of solid hulls in Banach lattices (submitted)*.**[11]**Joseph Diestel,*Sequences and series in Banach spaces*, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR**737004****[12]**G. Groenewegen,*On spaces of Banach lattice valued functions and measures (Thesis)*, Nijmegen University, Netherlands, 1982.**[13]**G. Groenewegen and A. van Rooij,*The modulus of a weakly compact operator*, Math. Z.**195**(1987), no. 4, 473–480. MR**900341**, 10.1007/BF01166700**[14]**Peter Meyer-Nieberg,*Banach lattices*, Universitext, Springer-Verlag, Berlin, 1991. MR**1128093****[15]**Constantin P. Niculescu,*Weak compactness in Banach lattices*, J. Operator Theory**6**(1981), no. 2, 217–231. MR**643692****[16]**Klaus D. Schmidt,*On the modulus of weakly compact operators and strongly additive vector measures*, Proc. Amer. Math. Soc.**102**(1988), no. 4, 862–866. MR**934857**, 10.1090/S0002-9939-1988-0934857-7**[17]**Anthony W. Wickstead,*Dedekind completeness of some lattices of compact operators*, Bull. Polish Acad. Sci. Math.**43**(1995), no. 4, 297–304 (1996). MR**1414786**

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Additional Information

**Z. L. Chen**

Affiliation:
Department of Applied Mathematics, Southwest Jiaotong University, Chengdu Sichuan 610031, People’s Republic of China

**A. W. Wickstead**

Affiliation:
Department of Pure Mathematics, The Queen’s University of Belfast, Belfast BT7 1NN, Northern Ireland

Email:
A. Wickstead@qub.ac.uk

DOI:
http://dx.doi.org/10.1090/S0002-9947-99-02431-9

Keywords:
Weakly compact operators,
Banach lattices

Received by editor(s):
May 7, 1997

Published electronically:
July 21, 1999

Article copyright:
© Copyright 1999
American Mathematical Society