The invariant subspace problem and Rosenblum operators I
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918062824882176 |
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| author | Fang, Junsheng Hou, Bingzhe Jiang, Chunlan Zhang, Yuanhang |
| author_facet | Fang, Junsheng Hou, Bingzhe Jiang, Chunlan Zhang, Yuanhang |
| contents | Let $T\in B(\mathcal{H})$ be an invertible operator. From the 1940's, Gelfand, Hille and Wermer investigated the invariant subspaces of $T$ by analyzing the growth of $\|T^n\|$, where $n\in \mathbb{Z}$. In this paper, we study the invariant subspaces of $T$ by estimating the growth of $\|T^n+λT^{-n}\|$, where $n\in \mathbb{N}$ and $λ$ is a nonzero complex constant. The key ingredient of our approach is introducing the notion of shift representation operators, which is based on the Rosenblum operators. In addition, by employing shift representation operators, we provide an equivalent of the Invariant Subspace Problem via the injectivity of certain Hankel operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15270 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The invariant subspace problem and Rosenblum operators I Fang, Junsheng Hou, Bingzhe Jiang, Chunlan Zhang, Yuanhang Functional Analysis Primary 47A15, 47B02, Secondary 47A16 Let $T\in B(\mathcal{H})$ be an invertible operator. From the 1940's, Gelfand, Hille and Wermer investigated the invariant subspaces of $T$ by analyzing the growth of $\|T^n\|$, where $n\in \mathbb{Z}$. In this paper, we study the invariant subspaces of $T$ by estimating the growth of $\|T^n+λT^{-n}\|$, where $n\in \mathbb{N}$ and $λ$ is a nonzero complex constant. The key ingredient of our approach is introducing the notion of shift representation operators, which is based on the Rosenblum operators. In addition, by employing shift representation operators, we provide an equivalent of the Invariant Subspace Problem via the injectivity of certain Hankel operators. |
| title | The invariant subspace problem and Rosenblum operators I |
| topic | Functional Analysis Primary 47A15, 47B02, Secondary 47A16 |
| url | https://arxiv.org/abs/2506.15270 |