On the hyperinvariant subspace problem, II
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Recently in  the question of whether every non-scalar operator on a complex Hilbert space H of dimension N0 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 showing 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 B u in (BCP) and C00 with ∥K∥ < ε has a nontrivial hyperinvariant subspace, then every nonscalar operator on H has a nontrivial hyperinvariant subspace. Indiana University Mathematics Journal ©,.
author list (cited authors)
Hamid, S. M., Onica, C., & Pearcy, C.