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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
DOI: https://doi.org/10.1090/S0002-9939-08-09775-X
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
MR Author ID: 775644
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

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.