Another special role of $ \mathrm{L}^\infty $-spaces-evolution equations and Lotz' theorem
In this paper, we review the underappreciated theorem by Lotz that tells us that every strongly continuous operator semigroup on a Grothendieck space with the Dunford-Pettis property is automatically uniformly continuous. A large class of spaces that carry these geometric properties are $ \mathrm{L}...
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| Format: | Article |
| Language: | English |
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AIMS Press
2024-12-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241716 |
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| author | Christian Budde |
| author_facet | Christian Budde |
| author_sort | Christian Budde |
| collection | DOAJ |
| description | In this paper, we review the underappreciated theorem by Lotz that tells us that every strongly continuous operator semigroup on a Grothendieck space with the Dunford-Pettis property is automatically uniformly continuous. A large class of spaces that carry these geometric properties are $ \mathrm{L}^\infty(\Omega, \Sigma, \mu) $ for non-negative measure spaces. This shows once again that $ \mathrm{L}^\infty $-spaces have to be treated differently. |
| format | Article |
| id | doaj-art-3a12e155fa6f4c718dd315bbef933bd6 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-3a12e155fa6f4c718dd315bbef933bd62025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912361583616610.3934/math.20241716Another special role of $ \mathrm{L}^\infty $-spaces-evolution equations and Lotz' theoremChristian Budde0University of the Free State, Department of Mathematics and Applied Mathematics, Faculty of Natural and Agriculture Sciences, PO Box 339, Bloemfontein 9300, South AfricaIn this paper, we review the underappreciated theorem by Lotz that tells us that every strongly continuous operator semigroup on a Grothendieck space with the Dunford-Pettis property is automatically uniformly continuous. A large class of spaces that carry these geometric properties are $ \mathrm{L}^\infty(\Omega, \Sigma, \mu) $ for non-negative measure spaces. This shows once again that $ \mathrm{L}^\infty $-spaces have to be treated differently.https://www.aimspress.com/article/doi/10.3934/math.20241716$ \mathrm{l}^\infty $-spaceslotz' theoremgrothendieck spacesdunford-pettis property |
| spellingShingle | Christian Budde Another special role of $ \mathrm{L}^\infty $-spaces-evolution equations and Lotz' theorem AIMS Mathematics $ \mathrm{l}^\infty $-spaces lotz' theorem grothendieck spaces dunford-pettis property |
| title | Another special role of $ \mathrm{L}^\infty $-spaces-evolution equations and Lotz' theorem |
| title_full | Another special role of $ \mathrm{L}^\infty $-spaces-evolution equations and Lotz' theorem |
| title_fullStr | Another special role of $ \mathrm{L}^\infty $-spaces-evolution equations and Lotz' theorem |
| title_full_unstemmed | Another special role of $ \mathrm{L}^\infty $-spaces-evolution equations and Lotz' theorem |
| title_short | Another special role of $ \mathrm{L}^\infty $-spaces-evolution equations and Lotz' theorem |
| title_sort | another special role of mathrm l infty spaces evolution equations and lotz theorem |
| topic | $ \mathrm{l}^\infty $-spaces lotz' theorem grothendieck spaces dunford-pettis property |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241716 |
| work_keys_str_mv | AT christianbudde anotherspecialroleofmathrmlinftyspacesevolutionequationsandlotztheorem |