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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 | References | Similar Articles | Additional Information

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 $T$.

References [Enhancements On Off] (What's this?)

  • [L] V. I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous operator, Funktsional. Anal. i Prilozhen. 7 (1973), No. 3, 55-56. (Russian)MR 54:8319
  • [R1] C. J. Read, A short proof concerning the invariant subspace problem, J. Lond. Math. Soc., (2) 33 (1986), 335-348. MR 87m:47020
  • [R2] C. J. Read, Quasinilpotent Operators and the Invariant Subspace Problem, J.Lond.Math.Soc., (2) 56 (1997), No. 3, 595-606. MR 98m:47004
  • [TV] V. G. Troitsky, On the modulus of C. J. Read's operator, Positivity 2 (1998), No. 3, 257-264. CMP 99:04

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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

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

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