ON THE MODULUS OF C. J. READ’S OPERATOR

Abstract. Let T: `1 → `1 be the quasinilpotent operator without an invariant subspace constructed by C. J. Read in [R3]. We prove that the modulus of this operator has an invari-ant subspace (and even an eigenvector). This answers a question posed by Y. Abramovich, C. Aliprantis and O. Burkinshaw in [AAB1, AAB3]. During the last several years there has been a noticeable increase of interest in the in-variant subspace problem for positive operators on Banach lattices. A rather complete and comprehensive survey on this topic is presented in [AAB3], to which we refer the reader for details and for an extensive bibliography. In particular, the following theorem was proved in [AAB1]. Theorem 1 ([AAB1, AAB3]). If the modulus of a continuous operator T: `p → `p (1 6 p < ∞) exists and is quasinilpotent, then T has a non-trivial closed invariant subspace which is an ideal. It follows that each positive quasinilpotent operator on `p (1 6 p < +∞) has a nontrivial closed invariant subspace. In the same papers the authors posed the following problem. Problem. Does every positive operator on `1 have an invariant subspace? Keeping in mind that each operator on `1 has a modulus and that C. J. Read in [R1, R2, R3] has constructed several operators on `1 without invariant subspaces, it was suggested in [AAB1, AAB3] that the modulus of some of these operators might be a natural candidate for a counterexample to the above problem. Following this suggestion, we will be dealing in this paper with the modulus of the quasinilpotent operator T constructed in [R3]. It turns out, quite surprisingly, that not only does |T | have an invariant subspace but it even has a positive eigenvector. This result increases the chances for an affirmative answer to the above problem. The paper is organized as follows. After introducing some necessary notation and termi-nology we prove a general theorem on the existence of an invariant subspace for the modulu