Hyperinvariant Subspace Problem for Quasinilpotent Operators

Abstract

In this paper we consider the hyperinvariant subspace problem for quasinilpotent operators. Let $$({\mathcal{CRQ}})$$ denote the class of quasinilpotent quasiaffinities Q in $${\mathcal{L}}(H)$$ such that Q * Q has an infinite dimensional reducing subspace M with Q * Q| M compact. It was known that if every quasinilpotent operator in $$({\mathcal{CRQ}})$$ 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 $$({\mathcal{CRQ}})$$ . The purpose of this paper is to provide sufficient conditions for elements in $$({\mathcal{CRQ}})$$ to have nontrivial hyperinvariant subspaces. We also introduce the notion of “stability” of extremal vectors to give partial solutions to the hyperinvariant subspace problem.