On the Hyperinvariant Subspace Problem. II

ABSTRACT. Recently in [6] the question of whether every non-scalar operator on a complex Hilbert space H of dimension @0 has a nontrivial hyperinvariant subspace was reduced to a special case; namely, the question whether every (BCP)-operator in C00 whose left essential spectrum is equal to some annulus centered at the origin has a nontrivial hyperinvariant subspace. In this note, we make additional contributions to this circle of ideas by show-ing that every (BCP)-operator in C00 is ampliation quasisimilar to a quasidiagonal (BCP)-operator in C00. Moreover, we show that there exists a fixed 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. 1