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

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77412We 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.5 pages, to appear in Proceedings of the AMSFunctional Analysis47A15Lomonosov's theorem cannot be extended to chains of four operatorstext