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

Troitsky, Vladimir

doi:10.1090/S0002-9939-99-05176-X

Lomonosov’s theorem cannot be extended to chains of four operators
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by Vladimir G. Troitsky

Proc. Amer. Math. Soc. 128 (2000), 521-525

DOI: https://doi.org/10.1090/S0002-9939-99-05176-X

Published electronically: June 24, 1999

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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 $T$.

References

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C. J. Read, Quasinilpotent operators and the invariant subspace problem, J. London Math. Soc. (2) 56 (1997), no. 3, 595–606. MR 1610408, DOI 10.1112/S0024610797005486
V. G. Troitsky, On the modulus of C. J. Read’s operator, Positivity 2 (1998), No. 3, 257–264.