Lomonosov's theorem cannot be extended to chains of four operators
Abstract
We show that the celebrated Lomonosov theorem cannot be improved by increasing the number of commuting operators. Specifically, we prove that if T is the operator on l1 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.
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