On the hyperinvariant subspace problem

In this paper, we introduce a new equivalence relation, ampliation quasisimilarity, on L(H), more general than quasisimilarity, that preserves the existence of nontrivial hyperinvariant subspaces. We show that if T does not have nontrivial hyperinvariant subspaces for elementary reasons, then T is ampliation quasisimilar to a (BCP)-operator in the class C00. This reduces the hyperinvariant subspace problem for operators in L(H) to a very special subcase of itself.