**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:*l*_{1}->*l*_{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.

**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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