Hyperinvariant subspaces of hyponormal operators: A constructive decomposition approach

It is shown that any hyponormal operator on an infinite-dimensional separable Hilbert space that admits a decomposition $T=R+V$, where $R$ is tridiagonal and $V$ is trace-class, has nontrivial closed hyperinvariant subspaces provided $T$ is not a multiple of the identity. We further discuss implications of this result for the invariant subspace problem of hyponormal operators answering, in particular, negatively to a question raised by Kim and Lee \cite{kimlee} regarding an explicit approach to such a problem. Finally, we characterize the existence of reducing subspaces for hyponormal operators addressing an approach by Aronszajn and Smith.