Normality of spaces of operators and quasi-lattices

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• 作者：Miek Messerschmidt
• 关键词：(Pre) ; ordered Banach space ; Operator norm ; Quasi ; lattice ; Normality ; Conormality ; Lorentz cone ; Primary 46B40 ; Secondary 47B60 ; 47H07 ; 46B42 ; 46A40
• 刊名：Positivity
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：19
• 期：4
• 页码：695-724
• 全文大小：600 KB
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  19.Robinson, D.W., Yamamuro, S.: The canonical half-norm, dual half-norms, and monotonic norms. Tohoku Math. J. (2) 35(3), 375鈥?86 (1983)MATH MathSciNet CrossRef
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• 作者单位：Miek Messerschmidt (1) (2)
  
  1. Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA, Leiden, The Netherlands
  2. Unit for BMI, North-West University, Private Bag X6001, Potchefstroom, 2520, South Africa
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Operator Theory
  Potential Theory
  Calculus of Variations and Optimal Control
  Econometrics
  
• 出版者：Birkh盲user Basel
• ISSN：1572-9281

文摘

We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces \(X\) and \(Y\) with closed cones we investigate normality of \(B(X,Y)\) in terms of normality and conormality of the underlying spaces \(X\) and \(Y\). 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 \(X\) and \(Y\) that are not Banach lattices, but for which \(B(X,Y)\) is normal. In particular, we show that a Hilbert space \(\mathcal {H}\) endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if \(\dim \mathcal {H}\ge 3\)), and satisfies an identity analogous to the elementary Banach lattice identity \(\Vert |x|\Vert =\Vert x\Vert \) which holds for all elements \(x\) 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. Keywords (Pre)-ordered Banach space Operator norm Quasi-lattice Normality Conormality Lorentz cone