Hyperinvariant subspaces of hyponormal operators: A constructive decomposition approach
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866913176348524544 |
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| author | Clemente, Norberto Gallardo-Gutiérrez, Eva A. |
| author_facet | Clemente, Norberto Gallardo-Gutiérrez, Eva A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_00865 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hyperinvariant subspaces of hyponormal operators: A constructive decomposition approach Clemente, Norberto Gallardo-Gutiérrez, Eva A. Functional Analysis 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. |
| title | Hyperinvariant subspaces of hyponormal operators: A constructive decomposition approach |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2606.00865 |