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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.16401 |
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| _version_ | 1866913486007697408 |
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| author | Kumar, Poornendu Rastogi, Shubham Tripathi, Raghavendra |
| author_facet | Kumar, Poornendu Rastogi, Shubham Tripathi, Raghavendra |
| contents | Douglas and Rudin proved that any unimodular function on the unit circle $\T$ can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of $\mathbb{C}^d$. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result [Proc. Lond. Math. Soc. 2009] on the approximation of matrix-valued unimodular functions on $\T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16401 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Douglas-Rudin Approximation theorem for operator-valued functions on the unit ball of $\mathbb{C}^d$ Kumar, Poornendu Rastogi, Shubham Tripathi, Raghavendra Functional Analysis 46E40, 32A99, 30J05 Douglas and Rudin proved that any unimodular function on the unit circle $\T$ can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of $\mathbb{C}^d$. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result [Proc. Lond. Math. Soc. 2009] on the approximation of matrix-valued unimodular functions on $\T$. |
| title | Douglas-Rudin Approximation theorem for operator-valued functions on the unit ball of $\mathbb{C}^d$ |
| topic | Functional Analysis 46E40, 32A99, 30J05 |
| url | https://arxiv.org/abs/2403.16401 |