# Normality of spaces of operators and quasi-lattices

Normality of spaces of operators and quasi-lattices We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces $$X$$ X and $$Y$$ Y with closed cones we investigate normality of $$B(X,Y)$$ B ( X , Y ) in terms of normality and conormality of the underlying spaces $$X$$ X and $$Y$$ Y . Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples $$X$$ X and $$Y$$ Y that are not Banach lattices, but for which $$B(X,Y)$$ B ( X , Y ) is normal. In particular, we show that a Hilbert space $$\mathcal {H}$$ H endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if $$\dim \mathcal {H}\ge 3$$ dim H ≥ 3 ), and satisfies an identity analogous to the elementary Banach lattice identity $$\Vert |x|\Vert =\Vert x\Vert$$ ‖ | x | ‖ = ‖ x ‖ which holds for all elements $$x$$ x of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Normality of spaces of operators and quasi-lattices

Positivity, Volume 19 (4) – Feb 4, 2015
30 pages

/lp/springer_journal/normality-of-spaces-of-operators-and-quasi-lattices-it2p8S9rKG
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-015-0323-y
Publisher site
See Article on Publisher Site

### Abstract

We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces $$X$$ X and $$Y$$ Y with closed cones we investigate normality of $$B(X,Y)$$ B ( X , Y ) in terms of normality and conormality of the underlying spaces $$X$$ X and $$Y$$ Y . Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples $$X$$ X and $$Y$$ Y that are not Banach lattices, but for which $$B(X,Y)$$ B ( X , Y ) is normal. In particular, we show that a Hilbert space $$\mathcal {H}$$ H endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if $$\dim \mathcal {H}\ge 3$$ dim H ≥ 3 ), and satisfies an identity analogous to the elementary Banach lattice identity $$\Vert |x|\Vert =\Vert x\Vert$$ ‖ | x | ‖ = ‖ x ‖ which holds for all elements $$x$$ x of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices.

### Journal

PositivitySpringer Journals

Published: Feb 4, 2015

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