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 |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241716 |
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| Summary: | 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. |
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| ISSN: | 2473-6988 |