On reflexive operator algebras

Let $S$ be a weakly closed algebra of operators in a Hilbert space $H$, containing a maximal commutative $*$-subalgebra $\mathfrak A$ of the algebra of all bounded linear operators in $H$. One investigates the problem of the reflexivity of $S$ (an operator algebra is said to be reflexive if it contains every operator for which all invariant subspaces of the algebra are invariant). It is proved that each of the following two conditions is sufficient for the reflexivity of $S$: a) the lattice of the invariant subspaces of $S$ is symmetric; b) the algebra $\mathfrak A$ is generated by minimal projectors.
One obtains other results too, referring to more general problems.
Bibliography: 4 titles.