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 A with a commutative subspace lattice is strictly cyclic iff Lat ⁡ ( A ) ⊥ \operatorname {Lat}{(A)^ \bot } contains a finite number of atoms and each nonzero element of Lat ⁡ ( A ) ⊥ \operatorname {Lat}{(A)^ \bot } contains an atom. This leads to a characterization of the n 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 A with a commutative subspace lattice is strongly strictly cyclic iff Lat ⁡ ( A ) \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 ] × { 0 , 1 } × ⋯ \{ 0,1] \times \{ 0,1\} \times \cdots which satisfy A.C.C. We also show that if Alg ⁡ ( L ) \operatorname {Alg}(\mathcal {L}) is strictly cyclic and L ⊆ \mathcal {L} \subseteq atomic m.a.s.a. then Alg ⁡ ( L ) \operatorname {Alg}(\mathcal {L}) contains a strictly cyclic operator.