STRICTLY CYCLIC OPERATOR-ALGEBRAS

We prove several results about the lattice of invariant subspaces of general strictly cyclic and strongly strictly cyclic operator algebras. A reflexive operator algebra A with a commutative subspace lattice is strictly cyclic iff Lat(A) perpendicular-to contains a finite number of atoms and each nonzero element of Lat(A) perpendicular-to contains an atom. This leads to a characterization of the n-strictly cyclic reflexive algebras with a commutative subspace lattice as well as an extensive generalization of D. A. Herrero's result that there are no triangular strictly cyclic operators. A reflexive operator algebra A with a commutative subspace lattice is strongly strictly cyclic iff Lat(A) satisfies A.C.C. The distributive lattices which are attainable by strongly strictly cyclic reflexive algebras are the complete sublattices of {0, 1] x {0, 1} x ... which satisfy A.C.C. We also show that if Alg(L) is strictly cyclic and L subset-or-is-equal-to atomic m.a.s.a. then Alg(L) contains a strictly cyclic operator.