Wiener-Hopf indices of unimodular functions on the imaginary axis
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913390754004992 |
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| author | Frazho, A. E. Ran, A. C. M. van Schagen, F. |
| author_facet | Frazho, A. E. Ran, A. C. M. van Schagen, F. |
| contents | This paper is concerned with the Wiener-Hopf indices of unimodular rational matrix functions on the imaginary axis. These indices play a role in the Fredholm theory for Wiener-Hopf integral operators. Our main result gives formulas for the Wiener-Hopf indices in terms of the matrices appearing in realizations of the factors in a Douglas-Shapiro-Shields factorization of the unimodular function. Two approaches to this problem are presented: one direct approach using operator theoretic methods, and a second approach using the Cayley transform which allows to use results for an analogous problem regarding unimodular functions on the unit circle and corresponding Toeplitz operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_09472 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wiener-Hopf indices of unimodular functions on the imaginary axis Frazho, A. E. Ran, A. C. M. van Schagen, F. Functional Analysis 47A68, 47B35, 47A56, 47A53 This paper is concerned with the Wiener-Hopf indices of unimodular rational matrix functions on the imaginary axis. These indices play a role in the Fredholm theory for Wiener-Hopf integral operators. Our main result gives formulas for the Wiener-Hopf indices in terms of the matrices appearing in realizations of the factors in a Douglas-Shapiro-Shields factorization of the unimodular function. Two approaches to this problem are presented: one direct approach using operator theoretic methods, and a second approach using the Cayley transform which allows to use results for an analogous problem regarding unimodular functions on the unit circle and corresponding Toeplitz operators. |
| title | Wiener-Hopf indices of unimodular functions on the imaginary axis |
| topic | Functional Analysis 47A68, 47B35, 47A56, 47A53 |
| url | https://arxiv.org/abs/2406.09472 |