Hyperinvariant subspaces of block-triangular operators on Hilbert space

Abstract

We show that if a nonscalar operator on a separable Hilbert space has a nontrivial invariant subspace, then it has also a nontrivial hyperinvariant subspace. Thus the hyperinvariant subspace problem is equivalent to the invariant subspace problem. As a consequence we obtain that every bilateral weighted shift has a proper hyprinvariant subspace. Our proof is based on a recent structure theorem in \cite{HP}, originated in the approach to almost invariant half-spaces in \cite{APTT} (see also \cite{Tc}). The idea that such a result would be possible came from the paper \cite{Pearcy} by the second author which was submitted to Acta Szeged for publication. The manuscript contained the construction we use herein and also set forth the rank inequality that we use to obtain the contradiction that yields the desired theorem. The proof given in \cite{Pearcy} of that theorem was incorrect, however, and it's proof turned out to be rather difficult, and was eventually found by the first author of this paper.