A geometric approach to the compressed shift operator on the Hardy space over the bidisk
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918288400842752 |
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| author | Lu, Yufeng Yang, Yixin Zu, Chao |
| author_facet | Lu, Yufeng Yang, Yixin Zu, Chao |
| contents | This paper studies the compressed shift operator $S_z$ on the Hardy space over the bidisk via the geometric approach. We calculate the spectrum and essential spectrum of $S_z$ on the Beurling type quotient modules induced by rational inner functions, and give a complete characterization for $S_z^*$ to be a Cowen-Douglas operator. Then we extend the concept of Cowen-Douglas operator to be the generalized Cowen-Douglas operator, and show that $S_z^*$ is a generalized Cowen-Douglas operator. Moreover, we establish the connection between the reducibility of the Hermitian holomorphic vector bundle induced by kernel spaces and the reducibility of the generalized Cowen-Douglas operator. By using the geometric approach, we study the reducing subspaces of $S_z$ on certain polynomial quotient modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09145 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A geometric approach to the compressed shift operator on the Hardy space over the bidisk Lu, Yufeng Yang, Yixin Zu, Chao Functional Analysis Complex Variables Primary 47B38, Secondary 47A15 This paper studies the compressed shift operator $S_z$ on the Hardy space over the bidisk via the geometric approach. We calculate the spectrum and essential spectrum of $S_z$ on the Beurling type quotient modules induced by rational inner functions, and give a complete characterization for $S_z^*$ to be a Cowen-Douglas operator. Then we extend the concept of Cowen-Douglas operator to be the generalized Cowen-Douglas operator, and show that $S_z^*$ is a generalized Cowen-Douglas operator. Moreover, we establish the connection between the reducibility of the Hermitian holomorphic vector bundle induced by kernel spaces and the reducibility of the generalized Cowen-Douglas operator. By using the geometric approach, we study the reducing subspaces of $S_z$ on certain polynomial quotient modules. |
| title | A geometric approach to the compressed shift operator on the Hardy space over the bidisk |
| topic | Functional Analysis Complex Variables Primary 47B38, Secondary 47A15 |
| url | https://arxiv.org/abs/2601.09145 |