Isometries with Isomorphic Invariant Subspace Lattices

J. B. Conway and T. A. Gillespie (J. Funct. Anal.64 (1985), 178–189) characterized those reductive normal operators which have isomorphic invariant subspace lattices. In a subsequent paper (J. Operator Theory22 (1989), 31–49) they gave several necessary conditions of isomorphism in the class of nonreductive isometries. In this paper, we provide a new necessary condition when the isometry contains a bilateral shift. Furthermore, we give complete characterization if the nonreductive components of the isometries are cyclic. It turns out that this characterization is of different types in the unitary and in the nonunitary case. We describe also when absolutely continuous unitary operators have spatially isomorphic invariant subspace lattices. Our results provide answers for questions posed in the second Conway and Gillespie paper referenced above.