Operator algebras and invariant subspaces lattices. I

Given a bounded linear operator $T$, we study the following questions: when the commutant $\{T\}'$ is commutative; when each operator in the bicomrautant $\{T\}''$ can be approximated by polynomials of $T$ in the weak operator topology, the problem of reflexivity and others. These questions are solved for some classes of operators.