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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/2409.20298 |
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| _version_ | 1866914106717503488 |
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| author | Aleman, Alexandru Richter, Stefan |
| author_facet | Aleman, Alexandru Richter, Stefan |
| contents | Let $D(μ)$ denote a harmonically weighted Dirichlet space on the unit disc $\mathbb D$. We show that outer functions $f\in D(μ)$ are cyclic in $D(μ)$, whenever $\log f$ belongs to the Pick-Smirnov class $N^+(D(μ))$. If $f$ has $H^\infty$-norm less than or equal to 1, then cyclicity can also be checked via iterated logarithms. For example, we show that such outer functions $f$ are cyclic, whenever $\log(1+ \log(1/f))\in N^+(D(μ))$. This condition can be checked by verifying that $\log(1+ \log(1/f))\in D(μ)$.
If $f$ satisfies a mild extra condition, then the conditions also become necessary for cyclicity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20298 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cyclicity and iterated logarithms in the Dirichlet space Aleman, Alexandru Richter, Stefan Functional Analysis Let $D(μ)$ denote a harmonically weighted Dirichlet space on the unit disc $\mathbb D$. We show that outer functions $f\in D(μ)$ are cyclic in $D(μ)$, whenever $\log f$ belongs to the Pick-Smirnov class $N^+(D(μ))$. If $f$ has $H^\infty$-norm less than or equal to 1, then cyclicity can also be checked via iterated logarithms. For example, we show that such outer functions $f$ are cyclic, whenever $\log(1+ \log(1/f))\in N^+(D(μ))$. This condition can be checked by verifying that $\log(1+ \log(1/f))\in D(μ)$. If $f$ satisfies a mild extra condition, then the conditions also become necessary for cyclicity. |
| title | Cyclicity and iterated logarithms in the Dirichlet space |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2409.20298 |