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Positivity pp 255–279Cite as

Regular Operators between Banach Lattices

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Part of the book series: Trends in Mathematics ((TM))

Abstract

If X and Y are Banach lattices then there are several spaces of linear operators between them that may be studied. \( \mathcal{L}^r \) (X, Y) is the space of all norm bounded operators from X into Y. There is no reason to expect there to be any connection between the order structure of X and Y and that of \( \mathcal{L} \) (X, Y). \( \mathcal{L}^r \) (X, Y) is the space of regular operators, i.e., the linear span of the positive operators. This at least has the merit that when it is ordered by the cone of positive operators then that cone is generating. \( \mathcal{L}^b \) (X, Y) is the space of order bounded operators, which are those that map order bounded sets in X to order bounded sets in Y. We always have \( \mathcal{L}^r (X,Y) \subseteq \mathcal{L}^b (X,Y) \subseteq \mathcal{L}(X,Y) \) and both inclusions may be proper.

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Authors and Affiliations

  1. Pure Mathematics Research Centre, Queen’s University Belfast, University Road, Belfast, Northern Ireland, BT7 1NN, UK

    A. W. Wickstead

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  1. A. W. Wickstead
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Editors and Affiliations

  1. Institut Préparatoire aux Etudes Scientifiques et Techniques, Université de Carthage, B.P. 51, 2070, La Marsa, Tunisia

    Karim Boulabiar

  2. Department of Mathematics, University of Mississippi, Hume Hall 305, P. O. Box 1848, University, MS, 38677-1848, USA

    Gerard Buskes

  3. Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 1060, Tunis, Tunisia

    Abdelmajid Triki

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Wickstead, A.W. (2007). Regular Operators between Banach Lattices. In: Boulabiar, K., Buskes, G., Triki, A. (eds) Positivity. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8478-4_9

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