On the structure of a class of operators

In this dissertation we study certain classes of operators on a separable, complex, in??nite dimensional Hilbert space H, speci??cally from the point of view of properties of the hyperlattice (i.e., lattice of hyperinvariant subspaces) for such operators. We show that every (BCP)-operator in C00 is hyperquasisimilar to a quasidiagonal (BCP)- operator in C00. Moreover we show that there exists a ??xed block diagonal (BCP)- operator Bu with the property that if every compact perturbation Bu + K of Bu in (BCP) and C00 with kKk < " has a nontrivial hyperinvariant subspace, then every nonscalar operator on H has a nontrivial hyperinvariant subspace. This shows that the study of the structure of the hyperlattice of an arbitrary operator on Hilbert space is essentially equivalent to the study of the hyperlattice structure of some much smaller, special classes of operators, and it is these on which we concentrate. Moreover, we study some special subclasses (B??) and (S??) of the class of in- vertible (BCP)-operators with a view of obtaining some insight into the problem of determining the structure of operators in these classes.