Abstract It is shown that any hyponormal operator on an infinite‐dimensional separable Hilbert space that admits a decomposition , where is tridiagonal and is trace‐class, has nontrivial closed hyperinvariant subspaces provided 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 Complex Anal. Oper. Theory 18 (2024), no. 4, Paper No. 100, 12 pp. 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.
Clemente et al. (Mon,) studied this question.