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A ξ-weak Grothendieck compactness principle

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  10 March 2021

KEVIN BEANLAND  and
RYAN M. CAUSEY
KEVIN BEANLAND
Affiliation:
Washington and Lee University, 204 W. Washington Street, Chavis Hall Room 103, Lexington, VA24450, U.S.A. e-mail: [email protected]
RYAN M. CAUSEY
Affiliation:
123 Bachelor Hall, 301 S Patterson Avenue, Oxford, OH45056, U.S.A. e-mail: [email protected]
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Abstract

For 0 ≤ ξ ≤ ω1, we define the notion of ξ-weakly precompact and ξ-weakly compact sets in Banach spaces and prove that a set is ξ-weakly precompact if and only if its weak closure is ξ-weakly compact. We prove a quantified version of Grothendieck’s compactness principle and the characterisation of Schur spaces obtained in [7] and [9]. For 0 ≤ ξ ≤ ω1, we prove that a Banach space X has the ξ-Schur property if and only if every ξ-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence. The ξ = 0 and ξ= ω1 cases of this theorem are the theorems of Grothendieck and [7], [9], respectively.

MSC classification

Primary: 46B03: Isomorphic theory (including renorming) of Banach spaces
Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 1 , January 2022 , pp. 231 - 246
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Argyros, S. A., Godefroy, G. and Rosenthal, H. P.. Descriptive set theory and Banach spaces. In Handbook of the Geometry of Banach Spaces, vol. 2, pages 1007–1069 (North-Holland, Amsterdam, 2003).CrossRefGoogle Scholar
Argyros, S. A., Mercourakis, S. and Tsarpalias, A.. Convex unconditionality and summability of weakly null sequences. Israel J. Math. 107 (1998), 157–193.CrossRefGoogle Scholar
Azimi, P. and Hagler, J. N.. Examples of hereditarily Banach spaces failing the Schur property. Pacific J. Math. 122(2) (1986), 287–297.CrossRefGoogle Scholar
Beanland, K., Causey, R., Freeman, D. and Wallis, B.. Classes of operators determined by ordinal indices. J. Funct. Anal. 271(6) (2016), 1691–1746.CrossRefGoogle Scholar
Causey, R. M.. Concerning the Szlenk index. Studia Math. 236(3) (2017), 201–244.CrossRefGoogle Scholar
Causey, R. M. and Navoyan, K. V.. ξ-completely continuous operators and ξ-Schur Banach spaces. J. Funct.Anal. 276(7) (2019), 2052–2102.CrossRefGoogle Scholar
Dowling, P. N., Freeman, D., Lennard, C. J., Odell, E., Randrianantoanina, B. and Turett, B.. A weak Grothendieckcompactness principle. J. Funct. Anal. 263(5) (2012), 1378–1381.CrossRefGoogle Scholar
Dowling, P. N. and Mupasiri, D.. Grothendieck compactness principle for the Mackey dual topology. J. Math. Anal. Appl. 410(1) (2014), 483–486.CrossRefGoogle Scholar
Johnson, W. B., Lillemets, R. and Oja, E.. Representing completely continuous operators through weakly ∞-compact operators. Bull. Lond. Math. Soc. 48(3) (2016), 452–456.CrossRefGoogle Scholar
Judd, R. and Odell, E.. Concerning Bourgain’s -index of a Banach space. Israel J. Math. 108 (1998), 145–171.CrossRefGoogle Scholar
Lopez-Abad, J., Ruiz, C. and Tradacete, P.. The convex hull of a Banach-Saks set. J. Funct. Anal. 266(4) (2014), 2251–2280.CrossRefGoogle Scholar