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

 

A version of Lomonosov's theorem for collections of positive operators


Authors: Alexey I. Popov and Vladimir G. Troitsky
Journal: Proc. Amer. Math. Soc. 137 (2009), 1793-1800
MSC (2000): Primary 47B65; Secondary 47A15
Published electronically: December 29, 2008
MathSciNet review: 2470839
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Abstract: It is known that for every Banach space $ X$ and every proper $ WOT$-closed subalgebra $ \mathcal A$ of $ L(X)$, if $ \mathcal A$ contains a compact operator, then it is not transitive; that is, there exist non-zero $ x\in X$ and $ f\in X^*$ such that $ \langle f,Tx\rangle=0$ for all $ T\in\mathcal A$. In the case of algebras of adjoint operators on a dual Banach space, V. Lomonosov extended this result as follows: without having a compact operator in the algebra, one has $ \bigl\lvert\langle f,Tx\rangle\bigr\rvert\le\lVert T_*\rVert_e$ for all $ T\in\mathcal A$. In this paper, we prove a similar extension of a result of R. Drnovšek. Specifically, we prove that if $ \mathcal C$ is a collection of positive adjoint operators on a Banach lattice $ X$ satisfying certain conditions, then there exist non-zero $ x\in X_+$ and $ f\in X^*_+$ such that $ \langle f,Tx\rangle\le\lVert T_*\rVert_e$ for all $ T\in\mathcal C$.


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Additional Information

Alexey I. Popov
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta,Edmonton, Alberta, T6G 2G1, Canada
Email: apopov@math.ualberta.ca

Vladimir G. Troitsky
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta,Edmonton, Alberta, T6G 2G1, Canada
Email: vtroitsky@math.ualberta.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09775-X
PII: S 0002-9939(08)09775-X
Keywords: Positive operator, adjoint operator, transitive algebra
Received by editor(s): July 22, 2008
Published electronically: December 29, 2008
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.