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 : '1 ! '1 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. All Banach spaces in this note are assumed to be infinite dimensional and separable; all operators are linear and bounded. We say that an operator is a non-scalar operator if it is not a multiple of the identity operator. One of the major results in the history of the Invariant Subspace Problem was obtained by V. Lomonosov in (L) 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. For our initial operator T we will take an operator without an invariant subspace on '1