Lomonosov's theorem cannot be extended to chains of four operators, Proceedings of the AMS 128 (2000), 521-525.

Brief review

One of the major results in the history of the Invariant Subspace Problem was obtained by V.Lomonosov, who proved that if an operator T on a Banach space commutes with another non-scalar operator S and S commutes with a non-zero compact operator K, then T has an invariant subspace. Motivated by their study of the Invariant Subspace Problem for positive operators on Banach lattices, Y.A.Abramovich and C.D.Aliprantis have asked recently whether or not Lomonosov's theorem can be extended to chains of four or more operators. The purpose of this note is to answer this question in the negative. Specifically, we prove that if T:l1->l1 is the operator without a non-trivial closed invariant subspace constructed by C.J. Read, then there are three operators S1, S2 and K (non-multiples of the identity) such that T commutes with S1, S1 commutes with S2, S2 commutes with K, and K is compact. It is also shown that the commutant of T contains only series of T.

A follow-up comment: There is a small error in Section 2. Namely, the statement that Q and Q-1 belong to F is not true. However, this error can be easily fixed, so that the main statement of Section 2 (Proposition 2) itself is true. I have found a much shorter and simpler proof of it, see Section 1.2 of my Ph.D. Thesis.

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