Strictly cyclic operator algebras

John Froelich
1991 Transactions of the American Mathematical Society  
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)± contains a finite number of atoms and each nonzero element of Lat(^)x contains an atom. This leads to a characterization of the «-strictly cyclic reflexive algebras with a commutative subspace lattice as well as an extensive generalization of D. A. Herrero's
more » ... f 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(^) 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(^) is strictly cyclic and S? C atomic m.a.s.a. then Algf-S") contains a strictly cyclic operator.
doi:10.1090/s0002-9947-1991-0989575-0 fatcat:parpwsvvyvbchlui4di6ta745q