Hyperinvariant subspace problem for quasinilpotent operators

In this paper we consider the hyperinvariant subspace problem for quasinilpotent operators. Let (CRQ) denote the class of quasinilpotent quasi-affinities Q in L(H) such that Q*Q has an infinite dimensional reducing subspace M with Q*Q vertical bar(M) compact. It was known that if every quasinilpotent operator in (CRQ) has a nontrivial hyperinvariant subspace, then every quasinilpotent operator has a nontrivial hyperinvariant subspace. Thus it suffices to solve the hyperinvariant subspace problem for elements in (CRQ). The purpose of this paper is to provide sufficient conditions for elements in (CRQ) to have nontrivial hyperinvariant subspaces. We also introduce the notion of "stability" of extremal vectors to give partial solutions to the hyperinvariant subspace problem.