Lomonosov's theorem cannot be extended to chains of four operators
  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 l_1 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. 
Vladimir G.Troitsky