Positivity pp 255-279 | Cite as

Regular Operators between Banach Lattices

  • A. W. Wickstead
Part of the Trends in Mathematics book series (TM)


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.


Vector Lattice Banach Lattice Riesz Space Order Interval Regular Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Birkhäuser Verlag AG2007 2007

Authors and Affiliations

  • A. W. Wickstead
    • 1
  1. 1.Pure Mathematics Research CentreQueen’s University BelfastBelfast, Northern IrelandUK

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