Cellular-indecomposable subnormal operators (Q793961)

This paper is a study of the lattice of invariant subspaces for a subnormal operator. The unilateral shift has the property that the intersection of nontrivial invariant subspaces is a nontrivial invariant subspace. The authors show that if any subnormal operator has this property, then there is a simply connected open set G such that the spectrum equals \(\bar G\) and the essential spectrum is \(\partial G\) with Fredholm index negative one in G. The proof is based upon the fact that pure subnormal operators have an abundance of invariant subspaces whose elements can be identified with analytic functions. This remarkable fact is established by the authors in their paper on the reflexivity of subnormal operators [J. Funct. Anal. 37, 271-301 (1980; Zbl 0435.47034)]. More recently they have conjectured that every cyclic subnormal operator having the intersection property above is quasisimilar to the unilateral shift.