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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Banach lattices with Levi norms


Author: Birol Altin
Journal: Proc. Amer. Math. Soc. 135 (2007), 1059-1063
MSC (2000): Primary 46A40; Secondary 47B65
Published electronically: September 18, 2006
MathSciNet review: 2262906
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Abstract: Schmidt proved that an operator $ T$ from a Banach lattice $ E$ into a Banach lattice $ G$ with property $ (P)$ is order bounded if and only if its adjoint is order bounded, and in this case $ T$ satisfies $ \left\Vert \left\vert T\right\vert \right\Vert =\left\Vert \left\vert T^{\prime }\right\vert \right\Vert $. In the present paper the result is generalized to Banach lattices with Levi-Fatou norm serving as range, and some characterizations of Banach lattices with a Levi norm are given. Moreover, some characterizations of Riesz spaces having property $ (b)$ are also obtained.


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Additional Information

Birol Altin
Affiliation: Department of Mathematics, Faculty of Science and Arts, Gazi University, 06500, Teknikokullar, Ankara, Turkey
Email: birola@gazi.edu.tr

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08536-4
PII: S 0002-9939(06)08536-4
Keywords: Banach lattices, Levi norm, property $(b),$ regular norm
Received by editor(s): January 25, 2005
Received by editor(s) in revised form: June 22, 2005, July 18, 2005, and October 25, 2005
Published electronically: September 18, 2006
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society