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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A version of Lomonosov’s theorem for collections of positive operators
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by Alexey I. Popov and Vladimir G. Troitsky PDF
Proc. Amer. Math. Soc. 137 (2009), 1793-1800 Request permission

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
  • Received by editor(s): July 22, 2008
  • Published electronically: December 29, 2008
  • Communicated by: Nigel J. Kalton
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2470839