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

 

Lomonosov's theorem cannot be extended
to chains of four operators


Author: Vladimir G. Troitsky
Journal: Proc. Amer. Math. Soc. 128 (2000), 521-525
MSC (1991): Primary 47A15
Published electronically: June 24, 1999
MathSciNet review: 1641129
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Abstract: We show that the celebrated Lomonosov theorem cannot be improved by increasing the number of commuting operators. Specifically, we prove that if $T\colon\ell _1\to \ell _1$ is the operator without a non-trivial closed invariant subspace constructed by C. J. Read, then there are three operators $S_1$, $S_2$ and $K$ (non-multiples of the identity) such that $T$ commutes with $S_1$, $S_1$ commutes with $S_2$, $S_2$ commutes with $K$, and $K$ is compact. It is also shown that the commutant of $T$ contains only series of $T$.


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Additional Information

Vladimir G. Troitsky
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green St., Urbana, Illinois 61801
Email: vladimir@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05176-X
PII: S 0002-9939(99)05176-X
Keywords: Invariant subspaces, commutant
Received by editor(s): March 31, 1998
Published electronically: June 24, 1999
Additional Notes: The author was supported in part by NSF Grant DMS 96-22454.
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society