On the hyperinvariant subspace problem. II Sami HamidConstantin OnicaCarl Pearcy Recently in \cite{FP} the question of whether every nonscalar operator on a complex Hilbert space $\mathcal{H}$ of dimension $\aleph_{0}$ has a nontrivial hyperinvariant subspace was reduced to a special case; namely, the question whether every (BCP)-operator in $C_{00}$ 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 showing that every (BCP)-operator in $C_{00}$ is ampliation quasisimilar to a quasidiagonal (BCP)-operator in $C_{00}$. Moreover, we show that there exists a fixed block diagonal (BCP)-operator $B_{u}$ with the property that if every compact perturbation $B_{u}+K$ of $B_{u}$ in (BCP) and $C_{00}$ with $\|K\|<\varepsilon$ has a nontrivial hyperinvariant subspace, then every nonscalar operator on $\mathcal{H}$ has a nontrivial hyperinvariant subspace. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2639 10.1512/iumj.2005.54.2639 en Indiana Univ. Math. J. 54 (2005) 743 - 754 state-of-the-art mathematics http://iumj.org/access/