The norm of the backward shift on $H^4$ is $\sqrt[4]φ$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910209134297088 |
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| author | Bampouras, Konstantinos Llinares, Adrián |
| author_facet | Bampouras, Konstantinos Llinares, Adrián |
| contents | We prove that the backward shift operator on $H^4$ has norm equal to $\sqrt[4]φ$, with $φ= \frac{1 + \sqrt{5}}{2}$. Furthermore, we characterize all extremal functions; they are precisely the functions of the form \[ f(z) = μ\left( I(z) - \sqrt{\frac{1}{2φ}}\right), \] where $μ\in \mathbb{C}$ and $I$ is an inner function with $I(0) = \sqrt{\fracφ{2}}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_10469 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The norm of the backward shift on $H^4$ is $\sqrt[4]φ$ Bampouras, Konstantinos Llinares, Adrián Complex Variables Classical Analysis and ODEs Functional Analysis 47B38 (Primary), 30H10, 30J05 (Secondary) We prove that the backward shift operator on $H^4$ has norm equal to $\sqrt[4]φ$, with $φ= \frac{1 + \sqrt{5}}{2}$. Furthermore, we characterize all extremal functions; they are precisely the functions of the form \[ f(z) = μ\left( I(z) - \sqrt{\frac{1}{2φ}}\right), \] where $μ\in \mathbb{C}$ and $I$ is an inner function with $I(0) = \sqrt{\fracφ{2}}$. |
| title | The norm of the backward shift on $H^4$ is $\sqrt[4]φ$ |
| topic | Complex Variables Classical Analysis and ODEs Functional Analysis 47B38 (Primary), 30H10, 30J05 (Secondary) |
| url | https://arxiv.org/abs/2605.10469 |