Extension Theorems and the Riesz Decomposition Property

Extension Theorems and the Riesz Decomposition Property Positivity 7: 87–93, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands. Extension Theorems and the Riesz Decomposition Property A Short Survey on the Results Obtained by the Authors in Recent Years ˘ ˘ NICOLAE DANET ¸ and RODICA-MIHAELA DANET ¸ Technical University of Civil Engineering of Bucharest, 122–124 Lacul Tei Blvd., Bucharest 38, 72302 Romania. E-mail: ndanet@fx.ro Mathematics Subject Classification (2000): Primary: 47B60; Secondary: 47B65, 46A22, 46B42. Key words: Banach lattices, Hahn–Banach Theorem, odered vector spaces, regular operators. 1. Some Definitions and Notations An ordered vector space F has the Riesz decomposition property (RDP in short) if for any positive elements z, x ,x with z  x + x , there exist z ,z such that 1 2 1 2 1 2 0  z  x , 0  z  x and z = z + z . The RDP is equivalent with the 1 1 2 2 1 2 finite interpolation property, i.e., if x ,x  z ,z then there exists y such that 1 2 1 2 x ,x  y  z ,z ([4], p. 129). Every Riesz space has the Riesz decomposition 1 2 1 2 property, but the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Extension Theorems and the Riesz Decomposition Property

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2003 by Kluwer Academic Publishers
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.
10.1023/A:1025883207672
Publisher site
See Article on Publisher Site

Abstract

Positivity 7: 87–93, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands. Extension Theorems and the Riesz Decomposition Property A Short Survey on the Results Obtained by the Authors in Recent Years ˘ ˘ NICOLAE DANET ¸ and RODICA-MIHAELA DANET ¸ Technical University of Civil Engineering of Bucharest, 122–124 Lacul Tei Blvd., Bucharest 38, 72302 Romania. E-mail: ndanet@fx.ro Mathematics Subject Classification (2000): Primary: 47B60; Secondary: 47B65, 46A22, 46B42. Key words: Banach lattices, Hahn–Banach Theorem, odered vector spaces, regular operators. 1. Some Definitions and Notations An ordered vector space F has the Riesz decomposition property (RDP in short) if for any positive elements z, x ,x with z  x + x , there exist z ,z such that 1 2 1 2 1 2 0  z  x , 0  z  x and z = z + z . The RDP is equivalent with the 1 1 2 2 1 2 finite interpolation property, i.e., if x ,x  z ,z then there exists y such that 1 2 1 2 x ,x  y  z ,z ([4], p. 129). Every Riesz space has the Riesz decomposition 1 2 1 2 property, but the

Journal

PositivitySpringer Journals

Published: Oct 17, 2004

References

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