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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.10086 |
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| _version_ | 1866912550961020928 |
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| author | yufang, Xie shanshan, Ji jing, Xu Kui, Ji |
| author_facet | yufang, Xie shanshan, Ji jing, Xu Kui, Ji |
| contents | Let $\mathcal{A}$ denote the operator class in which every nonzero intertwiner between two operators in $\mathcal{A}$ has dense range. Utilizing the operators in $\mathcal{A}$ as atoms and the flag structure as connection, we introduce an extended operator class $\mathcal{F}_{n}(\mathcal{A}) (n\in\mathbb{N}\ \mbox{and}\ n\ge2)$, along with its subclass $\mathcal{OF}_{n}(\mathcal{A})$. We establish that, under certain conditions, quasi-similarity within the classes $\mathcal{F}_{n}(\mathcal{A})$ and $\mathcal{OF}_{n}(\mathcal{A})$ is equivalent, which provides an approach to describing quasi-similarity and similarity for high-index Fredholm operators. Furthermore, we demonstrate that quasi-similarity implies similarity for a large number of operators in $\mathcal{F}_{n}(\mathcal{A})$, thereby yielding a partial answer to the question raised by D.A. Herrero and generalizing existing numerous results. As applications, several examples of quasi-similarity and similarity involving multiplication operators on vector-valued reproducing kernel Hilbert spaces are presented. Lastly, we show that the strong irreducibility is preserved up to quasi-similarity within the class $\mathcal{F}_{n}(\mathcal{A})$. This offers a partial solution to C.L. Jiang's question. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10086 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the quasi-similarity of operators with flag structure yufang, Xie shanshan, Ji jing, Xu Kui, Ji Functional Analysis Let $\mathcal{A}$ denote the operator class in which every nonzero intertwiner between two operators in $\mathcal{A}$ has dense range. Utilizing the operators in $\mathcal{A}$ as atoms and the flag structure as connection, we introduce an extended operator class $\mathcal{F}_{n}(\mathcal{A}) (n\in\mathbb{N}\ \mbox{and}\ n\ge2)$, along with its subclass $\mathcal{OF}_{n}(\mathcal{A})$. We establish that, under certain conditions, quasi-similarity within the classes $\mathcal{F}_{n}(\mathcal{A})$ and $\mathcal{OF}_{n}(\mathcal{A})$ is equivalent, which provides an approach to describing quasi-similarity and similarity for high-index Fredholm operators. Furthermore, we demonstrate that quasi-similarity implies similarity for a large number of operators in $\mathcal{F}_{n}(\mathcal{A})$, thereby yielding a partial answer to the question raised by D.A. Herrero and generalizing existing numerous results. As applications, several examples of quasi-similarity and similarity involving multiplication operators on vector-valued reproducing kernel Hilbert spaces are presented. Lastly, we show that the strong irreducibility is preserved up to quasi-similarity within the class $\mathcal{F}_{n}(\mathcal{A})$. This offers a partial solution to C.L. Jiang's question. |
| title | On the quasi-similarity of operators with flag structure |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2505.10086 |