$H^\infty$--functional calculus for generators of semigroups that admit lower bounds
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917439542919168 |
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| author | Haak, Benhard H. Kunstmann, Peer Chr. |
| author_facet | Haak, Benhard H. Kunstmann, Peer Chr. |
| contents | We study $C_0$-semigroups on UMD Banach spaces under the assumption that a single semigroup operator admits a lower bound. We establish boundedness of $H^\infty$ functional calculi for the negative generator of such semigroups. Our approach is based on a dilation argument: combining a recent construction due to Madani with transference results for groups on UMD spaces, we embed the semigroup into a $C_0$-group on a larger space and transfer functional calculus estimates back to the original generator. As a byproduct, we obtain quantitative exponential lower bounds for the semigroup. We also show that equivalences due to Batty and Geyer, valid in Hilbert spaces, fail in the general Banach space setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24296 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $H^\infty$--functional calculus for generators of semigroups that admit lower bounds Haak, Benhard H. Kunstmann, Peer Chr. Functional Analysis Primary: 47A60, 47D06, Secondary: 46B20, 47A10, 47B40, 47A20 We study $C_0$-semigroups on UMD Banach spaces under the assumption that a single semigroup operator admits a lower bound. We establish boundedness of $H^\infty$ functional calculi for the negative generator of such semigroups. Our approach is based on a dilation argument: combining a recent construction due to Madani with transference results for groups on UMD spaces, we embed the semigroup into a $C_0$-group on a larger space and transfer functional calculus estimates back to the original generator. As a byproduct, we obtain quantitative exponential lower bounds for the semigroup. We also show that equivalences due to Batty and Geyer, valid in Hilbert spaces, fail in the general Banach space setting. |
| title | $H^\infty$--functional calculus for generators of semigroups that admit lower bounds |
| topic | Functional Analysis Primary: 47A60, 47D06, Secondary: 46B20, 47A10, 47B40, 47A20 |
| url | https://arxiv.org/abs/2604.24296 |