Strictly cyclic operator algebras

Abstract:

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 $\operatorname {Lat}{(A)^ \bot }$ contains a finite number of atoms and each nonzero element of $\operatorname {Lat}{(A)^ \bot }$ 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 $\operatorname {Lat}(A)$ satisfies A.C.C. The distributive lattices which are attainable by strongly strictly cyclic reflexive algebras are the complete sublattices of $\{ 0,1] \times \{ 0,1\} \times \cdots$ which satisfy A.C.C. We also show that if $\operatorname {Alg}(\mathcal {L})$ is strictly cyclic and $\mathcal {L} \subseteq$ atomic m.a.s.a. then $\operatorname {Alg}(\mathcal {L})$ contains a strictly cyclic operator.