Operator Norm Limits of Order Continuous Operators

Operator Norm Limits of Order Continuous Operators Positivity (2005) 9:341–355 © Springer 2005 DOI 10.1007/s11117-005-8513-7 Operator Norm Limits of Order Continuous Operators 1 2 A. K. KITOVER and A. W. WICKSTEAD 1 2 Department of Mathematics, Philadelphia Community College, USA; Department of Pure Mathematics, Queens University Belfast, University Road, Belfast BT7 1NN, Northern Ireland, UK. E-mail: a.wickstead@queens-belfast.ac.uk Received 25 February 2005; Accepted 11 May 2005 Mathematics Subject Classification 1991: 46B42, 47B65 Key words: order continuous operators, norm limits 1. Introduction The interaction between the order structure and the operator norm in spaces of regular operators between Banach lattices is far from well under- stood and is, in any case, not always a good one. This is in stark contrast to the interaction with the regular norm. For example if Y is Dedekind complete then the space of all regular operators from X into Y , L (X, Y ), is a Banach lattice under the regular norm whilst under the operator norm it is even possible that T → T = 0 with T ⊥ T for all n ∈ N ([6], Theorem n n 3.1). In particular the collection of all order continuous operators from X into Y is then a band in L (X, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Operator Norm Limits of Order Continuous Operators

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2005 by Springer
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/s11117-005-8513-7
Publisher site
See Article on Publisher Site

Abstract

Positivity (2005) 9:341–355 © Springer 2005 DOI 10.1007/s11117-005-8513-7 Operator Norm Limits of Order Continuous Operators 1 2 A. K. KITOVER and A. W. WICKSTEAD 1 2 Department of Mathematics, Philadelphia Community College, USA; Department of Pure Mathematics, Queens University Belfast, University Road, Belfast BT7 1NN, Northern Ireland, UK. E-mail: a.wickstead@queens-belfast.ac.uk Received 25 February 2005; Accepted 11 May 2005 Mathematics Subject Classification 1991: 46B42, 47B65 Key words: order continuous operators, norm limits 1. Introduction The interaction between the order structure and the operator norm in spaces of regular operators between Banach lattices is far from well under- stood and is, in any case, not always a good one. This is in stark contrast to the interaction with the regular norm. For example if Y is Dedekind complete then the space of all regular operators from X into Y , L (X, Y ), is a Banach lattice under the regular norm whilst under the operator norm it is even possible that T → T = 0 with T ⊥ T for all n ∈ N ([6], Theorem n n 3.1). In particular the collection of all order continuous operators from X into Y is then a band in L (X,

Journal

PositivitySpringer Journals

Published: Jun 8, 2005

References

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